Teoría de números primos

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    introductory prime number theoryresources

    he distribution of prime numbers (elementary, and visually-oriented

    resentation)

    he prime number theorem (proof outline and additional notes)

    iemann's zeta function (links)

    he Riemann Hypothesis (links)

    -adic numbers and adeles

    . Rockmore, "Chance in the Primes"

    his excellent and thorough article is intended as a commentary toupplement the first half of a popular talk on the Riemann Hypothesisven by Peter Sarnak at a 1998 MSRI conference [a video recording isvailable here.]

    ow the complex zeros of Riemann's zeta function 'encode' the

    stribution of primes

    he functional equation of the Riemann zeta function and related issues

    . Bump's lecture notes on the distribution of the Riemann zeta function's

    ontrivial zeros

    ummary of Ilan Vardi's (excellent) "Introduction to Analytic Number

    heory"

    uotations regarding the mystery of the prime distribution

    rchive tutorial mystery new search home contact

    ntroductory prime number theory resources

    http://www.maths.ex.ac.uk/~mwatkins/zeta/bump-zeros.psmailto:[email protected]:[email protected]://www.maths.ex.ac.uk/~mwatkins/zeta/bump-zeros.ps
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    prime numberis a positive whole number greater than one which is divisiblenly by itself and one. The first few are shown above. If the definition doesnt

    mean much to you, think of prime numbers as follows:

    you are presented with a pile of 28 stones, you will eventually deduce that the

    le can be divided into 2 equal piles of 14, 4 equal piles of 7, 7 equal piles of 4,tc. However, if one more stone is added to the pile, creating a total of 29, youan spend as long as you like, but you will never be able to divide it into equalles (other than the trivial 29 piles of 1 stone). In this way, we see that 29 is arime number, whereas 28 is non-prime orcomposite.

    ll composites break down uniquely into a product of prime factors: i.e. 28 = 2 x 27. Note that 2 is the only even prime - all other even numbers are divisible by 2.is neither prime nor composite by convention.

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    Distribution of primes tutorial - step 1

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    Why is the numberone not prime?(from the Prime Pages' list of

    frequently asked questions)

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    The number one is far more special than a prime! It is the unit (the building

    block) of the positive integers, hence the only integer which merits its own

    existence axiom in Peano's axioms. It is the only multiplicative identity (1.a

    a.1 = a for all numbers a). It is the only perfect nth power for all positiveintegers n. It is the only positive integer with exactly one positive divisor.

    But it is not a prime. So why not? Below we give four answers, each moretechnical than its precursor.

    Answer One: By definition of prime!

    The definition is as follows.

    An integergreater than one is called a prime number if its onlypositive divisors (factors) are one and itself.

    Clearly one is left out, but this does not really address the question "why?"

    Answer Two: Because of the purpose of primes.

    The formal notion of primes was introduced by Euclid in his study ofperfec

    numbers (in his "geometry" classic The Elements). Euclid needed to know

    when an integer n factored into a product ofsmallerintegers (a nontrivially

    factorization), hence he was interested in those numbers which did not factoUsing the definition above he proved:

    The Fundamental Theorem of Arithmetic

    Every positive integer greater than one can be written uniquely as aproduct of primes, with the prime factors in the product written in ord

    of nondecreasing size.Here we find the most important use of primes: they are the unique building

    blocks of the multiplicative group of integers. In discussion of warfare youoften hear the phrase "divide and conquer." The same principle holds inmathematics. Many of the properties of an integer can be traced back to theproperties of its prime divisors, allowing us to divide the problem (literally)

    into smaller problems. The number one is useless in this regard because a =

    1.a = 1.1.a = ... That is, divisibility by one fails to provide us any informatio

    about a.

    Answer Three: Because one is a unit.

    Why is the number one not a prime?

    http://www.utm.edu/research/primes/index.htmlhttp://www.utm.edu/research/primes/notes/faq/index.htmlhttp://primes.utm.edu/http://primes.utm.edu/search/http://primes.utm.edu/largest.htmlhttp://primes.utm.edu/primes/http://primes.utm.edu/prove/http://primes.utm.edu/howmany.shtmlhttp://primes.utm.edu/mersenne/index.htmlhttp://primes.utm.edu/glossary/http://primes.utm.edu/curios/http://primes.utm.edu/lists/http://primes.utm.edu/notes/faq/http://groups.yahoo.com/group/primenumbers/http://primes.utm.edu/bios/index.phphttp://primes.utm.edu/links/http://primes.utm.edu/bios/submission.phphttp://primes.utm.edu/mersenne/index.html#theoremshttp://primes.utm.edu/mersenne/index.html#theoremshttp://primes.utm.edu/mersenne/index.html#theoremshttp://primes.utm.edu/mersenne/index.html#theoremshttp://primes.utm.edu/bios/submission.phphttp://primes.utm.edu/links/http://primes.utm.edu/bios/index.phphttp://groups.yahoo.com/group/primenumbers/http://primes.utm.edu/notes/faq/http://primes.utm.edu/lists/http://primes.utm.edu/curios/http://primes.utm.edu/glossary/http://primes.utm.edu/mersenne/index.htmlhttp://primes.utm.edu/howmany.shtmlhttp://primes.utm.edu/prove/http://primes.utm.edu/primes/http://primes.utm.edu/largest.htmlhttp://primes.utm.edu/search/http://primes.utm.edu/http://www.utm.edu/research/primes/notes/faq/index.htmlhttp://www.utm.edu/research/primes/index.html
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    the units (or divisors of unity). These are the elements (numbers) which haa multiplicative inverse. For example, in the usual integers there are two un{1, -1}. If we expand our purview to include the Gaussian integers {a+bi | ab are integers}, then we have four units {1, -1, i, -i}. In some number systemthere are infinitely many units.

    So indeed there was a time that many folks defined one to be a prime, but it the importance of units in modern mathematics that causes us to be much

    more careful with with the number one (and with primes).

    Answer Four: By the Generalized Definition ofPrime.

    (See also the technical note in The prime Glossary' definition).

    There was a time that many folks defined one to be a prime, but it is the

    importance of units and primes in modern mathematics that causes us to be

    much more careful with the number one (and with primes). When we onlyconsider the positive integers, the role of one as a unit is blurred with its role

    as an identity; however, as we look at other number rings (a technical term fsystems in which we can add, subtract and multiply), we see that the class ofunits is of fundamental importance and they must be found before we caneven define the notion of a prime. For example, here is how Borevich and

    Shafarevich define prime number in their classic text "Number Theory:"

    An elementp of the ring D, nonzero and not a unit, is calledprime if it can not be decomposed into factorsp=ab, neither of

    which is a unit in D.Sometimes numbers with this property are called irreducible and then the

    name prime is reserved for those numbers which when they divide a productab, must divide a or b (these classes are the same for the ordinaryintegers--but not always in more general systems). Nevertheless, the units a

    a necessary precursors to the primes, and one falls in the class of units, notprimes.

    Another prime page by Chris K. Caldwell

    Why is the number one not a prime?

    http://primes.utm.edu/glossary/page.php?sort=Primehttp://primes.utm.edu/http://www.utm.edu/~caldwell/http://www.utm.edu/~caldwell/http://primes.utm.edu/http://www.utm.edu/research/primes/index.htmlhttp://primes.utm.edu/glossary/page.php?sort=Prime
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    ere the sequence of primes is presented graphically in terms of a step function or

    ounting function which is traditionally denoted . (Note: this has nothing to doith the value =3.14159...)

    he height of the graph at horizontal positionxindicates the number of primes

    ss than or equal tox. Hence at each prime value ofxwe see a vertical jump of onenit.

    ote that the positions of primes constitute just about the most fundamental,

    arguable, nontrivial information available to our consciousness. This transcendsstory, culture, and opinion. It would appear to exist 'outside' space and time and yet

    o be accessible to any consciousness with some sense of repetition, rhythm, orounting. The explanation in the previous page involving piles of stones can be usedo communicate the concept of prime numbers without the use of spoken language,r to a young child.

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    Distribution of primes tutorial - step 2

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    y zooming out to see the distribution of primes within the first 100atural numbers, we see that the discrete step function is beginning touggest a curve.

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    Distribution of primes tutorial - step 3

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    ooming out by another factor of 10, the suggested curve becomes evenmore apparent. Zooming much further, we would expect to see thegranular" nature of the actual graph vanish into the pixelation of

    he screen.

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    Distribution of primes tutorial - step 4

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    ow zooming out by a factor of 50, we get the above graph. Senior Max Planck Institutemathematician Don Zagier, in his article "The first 50 million primes" [Mathematicalntelligencer, 0 (1977) 1-19] states:

    "For me, the smoothness with which this curve climbs is one of the most

    astonishing facts in mathematics."Note however that you are notlooking at a smooth curve. Sufficiently powerful

    magnification would reveal that it was made of unit steps. The smoothness to whichagier refers is smoothness in limit.)

    he juxtaposition of this property with the apparent 'randomness' of the individualositions of the primes creates a sort of tension which can be witnessed in many

    opular-mathematical accounts of the distribution of prime numbers. Adjectives such as

    surprising", "astonishing", "remarkable", "striking", "beautiful", "stunning" and

    breathtaking" have been used. Zagier captures this tension perfectly in the same article"There are two facts about the distribution of prime numbers of which Ihope to convince you so overwhelmingly that they will be permanentlyengraved in your hearts. The first is that, despite their simple definitionand role as the building blocks of the natural numbers, the primenumbers...grow like weeds among the natural numbers, seeming to obeyno other law than that of chance, and nobody can predict where the nextone will sprout. The second fact is even more astonishing, for it states

    just the opposite: that the prime numbers exhibit stunning regularity,

    that there are laws governing their behaviour, and that they obey theselaws with almost military precision."

    i t

    Distribution of primes tutorial - step 5

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    n 1896, de la Valee Poussin and Hadamard simultaneously proved what had beenuspected for several decades, and what is now known as theprime number

    heorem:

    n words, the (discontinuous) prime counting function is asymptoticto the

    smooth) logarithmic functionx/logx. This means that the ratio of tox/logxcan

    e made arbitrarily close to 1 by considering large enoughx. Hencex/logxprovidesn approximation of the number of primes less than or equal tox, and if we takeufficiently largexthis approximation can be made as accurate as we would likeproportionally speaking - very simply, as close to 100% accuracy as is desired).

    he original proofs of the prime number theorem suggested other, betterpproximations. In the above graph we see thatx/logx, despite being asymptotic to

    , is far from being the smooth function which suggests when we zoom out

    there is plenty of room for improving the approximation. These improvements turnut to be greatly revealing.

    >

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    he first improvement on x/log x we consider is the logarithmic integralunction Li(x). This is defined to be the area under the curve of the

    unction 1/log u between 2 andx, as illustrated in the lefthand figure.auss arrived at this from the empirical fact that the probability of findingprime number at an integer value near a very large number x is almostxactly1/logx.

    Hopital's rule can be used to show that the ratio ofx/logxto

    tends to 1 asxapproaches infinity. Thus we may

    se either expression as an approximation to in the statement of

    he prime number theorem.

    n the righthand figure we see that this function provides an excellent fito the function . As Zagier states, "within the accuracy of our

    cture, the two coincide exactly."

    >

    http://mathworld.wolfram.com/LogarithmicIntegral.htmlhttp://mathworld.wolfram.com/LogarithmicIntegral.htmlhttp://mathworld.wolfram.com/LogarithmicIntegral.htmlhttp://mathworld.wolfram.com/LogarithmicIntegral.html
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    agier goes on to state:

    "There is one more approximation which I would like tomention. Riemann's research on prime numbers suggeststhat the probability for a large numberxto be prime should beeven closer to 1/logxif one counted not only the primenumbers but also thepowers of primes, counting the squareof a prime as half a prime, the cube of a prime as a third, etc.This leads to the approximation

    or, equivalently,

    The function on the right side. . .is denoted by R(x), in honourof Riemann. It represents an amazingly good approximationto as the above values show."

    o be clear about this, it should be pointed out that the explicit definitionor the the function R(x) is

    here are the Mbius numbers. These are defined to be zero

    hen n is divisible by a square, and otherwise to equal (-1)kwhere kishe number of distinct prime factors in n. As 1 has no prime factors, it

    ollows that (1) = 1.

    Distribution of primes tutorial - step 8

    http://mathworld.wolfram.com/MoebiusFunction.htmlhttp://mathworld.wolfram.com/MoebiusFunction.html
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    seems, then, that the distribution of prime numbers 'points to' or impliesiemann's function R(x). This function can be thought of as a smootheal to which the actual, jagged, prime counting function clings. The nextyer of information contained in the primes can be seen above, which is

    he result of subtracting from R(x). This function relates directly to

    he local fluctuations of the density of primes from their mean density.

    n their article "Are prime numbers regularly ordered?", three Argentinian

    haos theorists considered this function, treated it as a 'signal', and

    alculated its Liapunov exponents. These are generally computed forgnals originating with physical phenomena, and allow one to decidehether or not the underlying mechanism is chaotic. The authorsonclude

    "...a regular pattern describing the prime numberdistribution cannot be found. Also, from a physical pointof view, we can say that any physical system whosedynamics is unknown but isomorphic to the primenumber distribution has a chaotic behaviour."

    physicist shown the above graph might naturally think to attempt aourier analysis - i.e. to see if this noisy signal can be decomposed into aumber of periodic sine-wave functions. In fact something very much likehis is possible. To understand how, we must look at the Riemann zeta

    unction.

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    Distribution of primes tutorial - step 9

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    uler proved that the infinite sum 1 + 1/2 + 1/3 + 1/4 + 1/5 + . . . does notake a finite value (unlike, say, 1 + 1/2 + 1/4 + 1/8 + . . . which sums to 2).owever, if we raise the denominators to higher powers, for example

    1 + 1/22 + 1/32 + 1/42+ . . . or 1 + 1/23 + 1/33 + 1/43 + . . .

    e get a computable, finite sum. As it is possible to extend the concept ofxponentiationxn from integern to rational powers, and eventually to alleal-valued exponents, we can consider how smallx>1 can be made so

    hat

    1 + 1/2x+ 1/3x+ 1/4x+ 1/5x+ . . .

    till converges to a finite sum. It turns out that this sum converges for allx1, which then suggests the (real valued) zeta function

    = 1 + 1/2x+ 1/3x+ 1/4x+ 1/5x+ . . .

    We know that (a) this is a well-defined smooth function forx>1, (b) that its

    alue approaches infinity asxapproaches 1 from above, and (c) that asxends to infinity, approaches 1 (all higher terms dwindle to

    othing). Euler demonstrated that this infinite sum could also bexpressed in terms of an infinite product using the sequence of primeumbers:

    = [1/(1-1/2x)] [1/(1-1/3x)][1/(1-1/5x)][1/(1-1/7x)]. . .

    ormally, we write

    Distribution of primes tutorial - step 10

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    lthough this identity may be initially surprising, it's not actually thatfficult to demonstrate. See for example K. Devlin's elementary historicalccount "How Euler discovered the zeta function".

    he zeta function, as simple as it is to define, has some remarkableroperties, some of which we shall now examine.

    >

    Distribution of primes tutorial - step 10

    http://www.maths.ex.ac.uk/~mwatkins/zeta/devlin.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/devlin.pdf
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    Are Prime Numbers Regularly Ordered?

    Z. Gamba, J. Hernando

    Departamento de Fisica, Comision Nacional de Energia Atomica, Av.Libertador 8250, 1429 Buenos Aires, Argentina

    andL. Romanelli

    CAERCEM, Julian Alvarez 1218, 1414 Buenos Aires, Argentina

    This article appeared in Physics Letters A145, no. 2,3 (2 April 1990),106-108.

    bstract: The form of the prime number distribution function hasithstood the efforts of all the mathematicians that have considered it.ere we address this problem with the tools of chaotic dynamics and find

    hat, from a physical point of view, this distribution function is chaotic.

    [Commentary]

    he article was not part of the authors' usual current of research.ernando's 12 year old daughter Leticia brought the issue to his attentionhilst doing her homework! As they had been working on certainhaos-related issues, it occurred to them to apply certain tests to therime distribution which are normally applied to physical systems studied

    y chaos theorists.

    he article begins:

    A classical and long standing problem in number theory is the behaviourf the prime number distribution [1].

    everal attempts to find a regular pattern for the prime distribution haveeen made in the past [2] and, to our knowledge, none of them wasuccessful.

    rom a strictly mathematical point of view, this problem was extensivelytudied and still remains unsolved. However, some statistical results haveeen obtained, e.g. the fractions of intervals which contains exactly krimes follow a Poisson distribution [3]."

    rom a physical point of view we thought that if we find that thisstribution is chaotic, some non-rigorous answers can be provided."

    hey go on to discuss the connection with quantum chaology, and thetudy of the Riemann zeta function by Berry, et.al. in the search for a

    model of quantum chaos [4], and a possible proof of the Riemann

    ypothesis, based on the "spectral conjecture" of Hilbert and Plya. The

    Z. Gamba, et. al. - Are prime numbers regularly ordered?

    mailto:[email protected]:[email protected]://www.phy.bris.ac.uk/staff/berry_mv.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.phy.bris.ac.uk/staff/berry_mv.htmlmailto:[email protected]:[email protected]
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    gure 3 shows the variation of the largest Liapunov exponent with theze of the analyzed succession. The authors observe that it isnequivocally positive in all the range and, after an initial increase, a wideateau is reached with a convergent value of 0.11 after nearly 20,000oints. They calculated this with the method of Eckmann, et. al. [9]. Iturns out that this method also gives a minimal embedding dimension ofor the unknown underlying classical dynamic system.

    he other Liapunov exponents are 0.00, -0.04, and -0.14 to give a sum of0.07. The authors safely conclude that their statistics are not sufficient toonclude reliably whether the system in question is conservative orssipative.

    o conclude:

    Therefore we can safely conclude that a regular pattern describing therime number distribution cannot be found. Also, from a physical point ofew, we can say that any physical system whose dynamics is unknownut isomorphic to the prime number distribution has a chaotic behaviour."

    eferences:

    ] M.R. Schroeder, Number theory in science and communications,pringer series in information sciences (Springer, Berlin, 1986).

    2] M. Gardner, Scientific American, 210(3) (1964) 120.3] P.X. Gallagher, Mathematika23 (1976) 4.

    4] M.V. Berry, Proceedings of the Royal Society A413 (1987) 183; inpringer lecture notes in physics, Vol. 263. Quantum chaos andtatistical nuclear physics, eds. T.H. Seligman and H. Nishioka (Springer,erlin, 1986) p.1.

    6] I. Percival and F. Vivaldi, Physica D25 (1987) 105.

    7] J.-P. Eckmann and D. Ruelle, Rev. Mod. Physics57 (1985) 617; H.G.chuster, Deterministic Chaos (Physik-Verlag, Weinheim, 1984).

    9] J.-P. Eckmann, S. Oliffson Kamphorst, D. Ruelle and S. Ciliberto,hysical Review A34(1986) 4971.

    rchive tutorial mystery new search home contact

    Z. Gamba, et. al. - Are prime numbers regularly ordered?

    mailto:[email protected]:[email protected]
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    he function , whose graph is seen above, can be easily showno equal the infinite sum

    here is the Mbius Function which we encountered earlier. Recall

    hat it is defined on the natural numbers as follows:

    equals zero when n is divisible by a square, and otherwise equals

    1)kwhere kis the number ofdistinctprime factors in n.

    or example,

    (28) = 0, as 28 is divisible by 4 = 22

    (42) = (-1)3 = -1 as 42 = 2 x 3 x 7

    (55) = (-1)2 = 1 as 55 = 5 x 11.

    (242) = 0, as 242 is divisible by 121 = 112.

    n this way, acts as a "generating function" for the arithmetic

    formation associated with the function . Other functions

    onstructed from have similar properties. Three examples:

    ) generates the sequence of values as follows:

    Distribution of primes tutorial - step 11

    http://mathworld.wolfram.com/MoebiusFunction.htmlhttp://mathworld.wolfram.com/MoebiusFunction.html
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    ) log generates the sequence of values {l(n)} where l(n) is defined

    o be 1/kwhen n =pkand to be zero otherwise:

    log = l(1)/1x + l(2)/2x + l(3)/3x + . . .

    i) generates the sequence of values where

    (the von Mangoldt function) is zero unless n is a power of a prime

    in which case it takes the value logp:

    = (1)/1x + (2)/2x + (3)/3x + . . .

    >

    Distribution of primes tutorial - step 11

    http://mathworld.wolfram.com/MangoldtFunction.htmlhttp://mathworld.wolfram.com/MangoldtFunction.html
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    he zeta function we have encountered until now has only been definedor real values greater than 1.

    he real number line is contained within a larger number system, theane ofcomplex numbers of the form z= a + ib where a and b are reals.

    ere i is the imaginary unit, defined to be the square root of -1 which is a

    roperty not possessed by any real number. The complex plane

    epresented above has a real axis consisting of all complex numbers zith imaginary part zero, and an imaginary axis consisting of all complex

    umbers zwith real part zero.

    he darker portion of the real axis shows the domain of definition for theeal-valued zeta function. By a very important process called analytic

    ontinuation, the zeta function can be extended in a unique wayto be

    efined over the entire complex plane, with the exception of the point z= where it effectively becomes infinite. This point is known as a simple

    ole of the (analytically continued) Riemann zeta function.

    his newly extended zeta function is not only defined on complex values,ut is also complex-valued, i.e. is not generally a real number. For

    his reason, it is not easy to display its behaviour with a static graph. It

    maps points in to other points in , so it is perhaps best to try tomagine it in more 'dynamic' terms. Just as the real-valued zeta functionssentially maps the (real) intervalx> 1 onto itself by "inverting" it in aery particular and important way, the complex-valued Riemann zetaunction can be thought of roughly as "turning the complex planeside-out" in a very particular andprofoundlyimportant way. Thisurning inside-out" mapping is far from simple, as we shall go on to see.

    Distribution of primes tutorial - step 12

    http://mathworld.wolfram.com/ComplexNumber.htmlhttp://mathworld.wolfram.com/ComplexPlane.htmlhttp://mathworld.wolfram.com/AnalyticContinuation.htmlhttp://mathworld.wolfram.com/AnalyticContinuation.htmlhttp://mathworld.wolfram.com/AnalyticContinuation.htmlhttp://mathworld.wolfram.com/AnalyticContinuation.htmlhttp://mathworld.wolfram.com/ComplexPlane.htmlhttp://mathworld.wolfram.com/ComplexNumber.html
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    ere we see the modulus of the zeta function plotted over the complex

    ane where z=x+ iy. Recall the definition |a + ib|2

    = |a|2

    + |b|2

    . Note the pole at= 1.

    nfortunately we cannot represent the behaviour of itself with a static graph,

    ecause it is a complex-valued function over the complex plane. It may be usefulo think of it in more 'active' terms, acting on the complex plane to transport theoint zto the point .

    owever, a Mathematica application developed by Bernd Thaller(University of

    raz) allows 2D or 3D representation of complex-valued functions through these of colours. Here is a representation of the zeta function produced by Alex

    stashyn using this application.

    >

    Distribution of primes tutorial - step 13

    http://www.kfunigraz.ac.at/imawww/vqm/pages/colormap.htmlhttp://www.kfunigraz.ac.at/imawww/vqm/pages/colormap.htmlhttp://www.kfunigraz.ac.at/imawww/vqm/pages/colormap.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/colourzetaimage.jpghttp://www.maths.ex.ac.uk/~mwatkins/zeta/colourzetaimage.jpghttp://www.kfunigraz.ac.at/imawww/vqm/pages/colormap.html
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    his graph shows the behaviour of .

    his allows us to see some points zwhere is zero. These lie at the bases of

    he vertical spikes where becomes infinite. It appears that there is a row

    f such zeros running parallel, and close, to the imaginary axis. These turn out toe of great importance to the distribution of prime numbers.

    >

    Distribution of primes tutorial - step 14

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    he zeros of , that is the points in the complex plane where , are of

    articular interest.

    hey fall into two categories. The trivial zeros are the negative even integers2,-4,-6,-8,..., so-called because it is relatively easy to demonstrate that for

    atural numbers n. The nontrivial zeros are much more mysterious.

    iemann was able to prove that they must all lie in the vertical strip defined by 0

    Distribution of primes tutorial - step 15

    http://www.research.att.com/~amo/zeta_tableshttp://www.research.att.com/~amo/zeta_tables
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    ere we see the behaviour of the Riemann zeta function on the critical

    ne Re[z] = 1/2. Recall that all known zeros lie on this line. These graphshow parts of the image of this line after it has been mapped into by .

    he nontrivial zeros correspond to the points on the curve where itasses through the origin.

    hese graphs are taken from the article "Phase of the Riemann zeta

    unction and the inverted harmonic oscillator" by R.K. Bhaduri, Avinash

    hare and J. Law in which they state

    The loop structure of the function at = 1/2, with some near-circular

    hapes, is reminiscent of the Argand plots for the scattering amplitudes ofifferent partial waves in the analysis of resonances, for example inion-nucleon scattering."

    his seems to be more than a superficial resemblance, as the authors gon to argue that "The smooth phase of the function along the line of the

    eros is related to the quantum density of states of an inverted oscillator."his is just one of a number ofsurprising connections between physics

    nd prime numbers (in this case via the zeta function).

    >

    Distribution of primes tutorial - step 16

    http://www.maths.ex.ac.uk/~mwatkins/zeta/surprising.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/surprising.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/surprising.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/surprising.htm
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    We encountered this graph earlier. Recall that it shows us the deviationsf the prime counting function from the smooth approximating

    unction R(x). It was hinted that this noisy function might somehowecompose into fairly simple component functions. Indeed, this is thease.

    he usual process of Fourier analysis essentially decomposes "signals"uch as this into (periodic) sine wave functions. In this case, theomponent functions are quasi-periodic, based on sine waves but with a

    articular kind of logarithmic deformation.

    emarkably, the functions in question, the sum of which produces theunction seen above, are intimately connected with the nontrivial zeros ofhe zeta function which we've just seen.

    >

    Distribution of primes tutorial - step 17

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    he difference function R(x) - seen earlier can be expressed as the infinite

    um over the set of zeros (both trivial and nontrivial) of the Riemann zeta

    unction:

    his sum separates into sums over the trivial and nontrivial zeros respectively. Theormer is the relatively simple function R(x-2) + R(x-4) + R(x-6) + ...

    he sum over the nontrivial zeros can be expressed as the sum of the sequence ofunctions {-Tk(x)} where Tkis defined as follows:

    here the and are the kth pair of nontrivial zeta zeros, which we know

    must be complex conjugates. The first four functions T1(x), T2(x),T3(x), and T4(x)re pictured above.

    ur first apparent obstacle is that and are complex numbers. However, the

    unctionxkcan be meaningfully extended from real kto complex kin a fairlytraightforward way. This means that the and are also complex- valued.

    his also initially seems like a problem, as the Riemann function R defined earliers an approximation of was clearly intended to act on real values only.

    owever, by the same process of analytic continuation discussed earlier, R can bet d d t th ti l l t ki th f i b th G i

    Distribution of primes tutorial - step 18

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    we define the sequence of functions:

    hen from what we have seen, this should approach in limit as n

    ends to infinity. This is indeed what happens. Above we see R10(x) and

    29(x), which clearly appear to be progressing toward a step function. An

    nimation depicting this progression can be found here.

    ere we have an infinite sequence of smooth functions, each built upom the simple Riemann function R(x) and a finite number of theuasi-sinusoidal functions Tn(x) which we have just defined. The limit of

    his sequence is a discontinuous function, the familiar .

    n this way we see how the distribution of prime numbers, asharacterised by the counting function , can be reconstructed from

    he nontrivial zeros of the Riemann zeta function in the complex plane.he function R(x) deals with the average behaviour of , whereas the

    um of the Tk(x) captures the local fluctuations in the distribution. For this

    eason the sequence of nontrivial zeta zeros is sometimes described as

    eing "dual" to the sequence of prime numbers. They appear to be twospects of the same thing.

    Distribution of primes tutorial - step 19

    http://www.maths.ex.ac.uk/~mwatkins/zeta/pianim.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/pianim.htm
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    his graph compares the two approximate functions R10(x) and R29(x)

    ith the prime counting function . The fairly rapid convergence ofhe Rn(x) thus becomes apparent.

    >

    Distribution of primes tutorial - step 20

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    he distribution of nontrivial zeros of the zeta function is governed by theollowing asymptotic law:

    his gives the approximate number of zeros in the critical strip whosemaginary parts are between 0 and T. Using this, we can 'renormalise' thecations of the zeros, so that the average spacing between themecomes 1, and then study the statistics of the local fluctuations.

    has been hypothesised that these statistics correspond to the

    genvalue spacing statistics of a particular class of large randommatrices called the Gaussian Unitary Ensemble. GUE statistics are

    mportant because they relate to the energy spectra of particular systemstudied by quantum chaologists.

    his hypothesis is backed up by overwhelming numerical evidence,volving thousands of hours of supercomputer calculations carried out by. Odlyzko, among others. This makes plausible the spectral

    terpretation of the Riemann zeta function, which suggests that the

    maginary parts ('heights') of the nontrivial zeros correspond to thepectrum of eigenvalues of some operator. The spectral interpretation ismong the most promising approaches to proving the Riemann

    Distribution of primes tutorial - step 21

    http://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htm
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    eter Sarnak of Princeton University says, "Odlyzko's computations arehe first phenomenological insight that the zeros are absolutely,ndoubtedly, 'spectral' in nature. Riemann himself would be

    mpressed...At the phenomenological level, this is perhaps the mosttriking discovery about zeta since Riemann."

    emarkably, on the basis of his conjecture, and results from quantumhaology, Michael Berry has been able to predict certain subtle statistical

    ends in the spacing of the nontrivial zeros. These concern theirnumberariance, and have also been confirmed by Odlyzko's calculations. Abovee see a graphs displaying how accurately Berry's prediction (the curve)ts the actual data based on the spacings of the zeros (the small blackquares).

    ore information on this quantum chaological approach to proving theiemann Hypothesis can be found here.

    >

    Distribution of primes tutorial - step 21

    http://www.phy.bris.ac.uk/staff/berry_mv.htmlhttp://www.phy.bris.ac.uk/staff/berry_mv.html
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    sometimes have the feeling that the number system is comparable with theniverse that the astronomer is studying...The number system is something like aosmos."

    M. Jutila[quoted in K. Sabbagh, "Beautiful Mathematics", Prospect(Jan. 2002)]

    n recent years, a rapidly expanding body of work has been making unexpected,eemingly unrelated connections between the mysterious distribution of prime

    umbers and various branches of physics. Note that in general, mathematicsnforms' physics, but not vice versa. That is, mathematicians have traditionally beenble to provide physicists with useful insights and techniques, but this has been

    rgely a one-way process. What we are considering here is the reverse process,here insights and techniques derived from physics are shedding new light on puremathematical (in particular, number theoretical) concerns. The following pages aren attempt to document and archive this material as comprehensively as possible:

    number theory and physics archive

    s far as I am aware, no generalexplanation has been put forward as to why thishould be happening - i.e. why elaborate concepts, structures and phenomenaeveloped and studied by physicists, such as thermodynamic partition functions,uantum harmonic oscillators, spontaneous symmetry breaking, 1/f noise,agedorn catastrophes, pion-nucleon scattering, The Fokker-Planck equation, the

    Wiener-Khintchine duality relation, etc. should be so directly relevant to the purest ofure mathematical structures - the sequence of prime numbers.

    owever, some months before I became aware of any of the various materialompiled in the above-mentioned archive, an image emerged out of myream consciousness and turned into one of the strangest ideas ever to have

    mysterious occurences on the interface of physics and number theory

    http://www.utu.fi/ml/matlts/staff/jutila.htmhttp://www.prospect-magazine.co.uk/ArticleView.asp?Accessible=yes&P_Article=8406http://www.prospect-magazine.co.uk/ArticleView.asp?Accessible=yes&P_Article=8406http://www.utu.fi/ml/matlts/staff/jutila.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/pianim.htm
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    n s ome previously u nexplored con text , the fami l iar 'shape' of the sequence of

    r ime num bers is the resul t of a kind of d ynamic or evolut ion ary process.

    lthough I was aware that this might be completely meaningless, the idea had suchprofound effect on me that I attempted to find a rigorous mathematical framework which it could perhaps be given some meaning.

    n the process, I gradually discovered the various material mentioned above, andound it rather encouraging. For suppose we imagine that there is some meaningfulontent to my unorthodox speculation, i.e there does exist some sort of mysteriousynamics underlying the distribution of prime numbers. We might then expect to findertain 'clues' as to the nature of the dynamics hidden in the subtleties of the primestribution. Perhaps this is what is happening.

    he following page documents an initial attempt to provide a coherent mathematicaloundation for the concept. Although I am no longer sure what to think about thisea, I feel that it is worth leaving the document online for anyone who might be

    terested.

    his may turn out to be of more psychological than mathematical interest. I feel thatt the very least I picked up on some kind of 'resonance' with an important set ofeas emerging in the mathematical sciences, even if the form into which my mindanslated this may have been somewhat naive.

    'prime evolution' notes

    he following rather extraordinary preprint, which I discovered recently (April 2001),may turn out to be relevant here:

    V. Volovich, "Number theory as the ultimate physical theory", Preprint CERN-TH

    7 4781-4786 (1987)

    ou are invited to send me any comments, questions, or criticism. My original hopeas that the notes would eventually attract the attention of researchersppropriately specialised to either make use of the ideas involved, or toemonstrate conclusively that they are meaningless.

    mysterious occurences on the interface of physics and number theory

    http://www.maths.ex.ac.uk/~mwatkins/zeta/evolutionnotes.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/volovich1.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/volovich1.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/evolutionnotes.htm
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    his animation depicts the approximation of the prime counting function

    shown in blue) using the first 70 pairs of nontrivial zeros of the Riemann zeta

    unction in a variant ofvon Mangoldt's explicit formula. At each step, the current

    unction (shown in yellow) is modified by adding a waveform whose frequency and

    mplitude is related to the next pair of nontrivial (complex) zeros in a very simplend direct way.

    he horizontal animation at the top of this page is based on the slope of the yellowurve, so that the primes emerge from a homogeneous field as 'points of light'. It ishe closest thing I have managed to find within existing mathematical theory to myforementioned 'inner perception'.

    lso available is a similar animation depicting the emergence ofChebyshev's

    unction

    mysterious occurences on the interface of physics and number theory

    http://www.maths.ex.ac.uk/~mwatkins/zeta/pntproof.htm#explformhttp://www.maths.ex.ac.uk/~mwatkins/zeta/psianim.htmhttp://mathworld.wolfram.com/ChebyshevFunctions.htmlhttp://mathworld.wolfram.com/ChebyshevFunctions.htmlhttp://mathworld.wolfram.com/ChebyshevFunctions.htmlhttp://mathworld.wolfram.com/ChebyshevFunctions.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/psianim.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/pntproof.htm#explform
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    hich is a logarithmically-weighted prime counting function of great importance (forxample in the proof of the prime number theorem.)

    [graphics kindly produced by R. Manzoni on request]

    archive tutorial mystery new search home contact

    mysterious occurences on the interface of physics and number theory

    mailto:[email protected]:[email protected]:[email protected]:[email protected]
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    quantum mechanicsthe nontrivial Riemann zeta zeros interpreted as a spectrum of energy levels

    The Riemann zeta function is of interest to pure mathematiciansecause of its connection with prime numbers, but it is also a hugely

    mportant tool in quantum chaos because many calculations involving theiemann zeta function mirror the most fundamental manipulations inemiclassical work, those concerning the energy eigenvalues ofemiclassical systems and the action of the periodic orbits of thoseystems. Whereas the semiclassical calculations involve sums overeriodic orbits of the system in question...the Riemann zeta function

    ersion contains sums over prime numbers. As much knowledge has builtp about prime numbers over the years, the Riemann zeta calculationsre often more tractable than the periodic orbit ones, and so can providesight as to how the semiclassical calculations ought to proceed."

    om N. Snaith's Ph.D. thesis "Random matrix theory and zeta functions"

    University of Bristol, 2000)

    There's been an explosion of activity in this field - the progress in the last

    alf dozen years because of this marriage of these two fields has beenbsolutely incredible."

    . Gonek, Dr. Riemann's Zeros (Atlantic, 2002), p. 148

    One idea for proving the Riemann hypothesis is to give a spectral

    terpretation of the zeros. That is, if the zeros can be interpreted as the

    genvalues of 1/2 + iT, where Tis a Hermitian operator on some Hilbertpace, then since the eigenvalues of a Hermitian operator are real, theiemann hypothesis follows. This idea was originally put forth by Plya

    nd Hilbert, and serious support for this idea was found in the

    esemblance between the "explicit formulae" of prime number theory,hich go back to Riemann and von Mangoldt, but which were formalized

    s a duality principle by Weil, on the one hand, and the Selberg trace

    ormula on the other.

    he best evidence for the spectral interpretation comes from the theory ofhe Gaussian Unitary Ensemble (GUE), which show that the local

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://www.maths.bris.ac.uk/~mancs/http://www.maths.ex.ac.uk/~mwatkins/zeta/snaith-thesis.pdfhttp://www.amazon.co.uk/exec/obidos/ASIN/1843541009/ref%3Dbr_ed_top5_1_5/026-0190221-6772420http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Polya.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hilbert.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Riemann.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/weilexplicitformula.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/weilexplicitformula.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/physics4.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/physics4.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/bump-gue.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/bump-gue.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/physics4.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/physics4.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/weilexplicitformula.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/weilexplicitformula.htmhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Riemann.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hilbert.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Polya.htmlhttp://www.amazon.co.uk/exec/obidos/ASIN/1843541009/ref%3Dbr_ed_top5_1_5/026-0190221-6772420http://www.maths.ex.ac.uk/~mwatkins/zeta/snaith-thesis.pdfhttp://www.maths.bris.ac.uk/~mancs/
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    Gutzwiller gave a trace formula in the setting of quantum chaos whichelates the classical and quantum mechanical pictures. Given a chaoticclassical) dynamical system, there will exist a dense set of periodicrbits, and one side of the trace formula will be a sum over the lengths ofhese orbits. On the other side will be a sum over the eigenvalues of theamiltonian in the quantum-mechanical analog of the given classicalynamical system.

    his setup resembles the explicit formulas of prime number theory. In thisnalogy, the lengths of the prime periodic orbits play the role of theational primes, while the eigenvalues of the Hamiltonian play the role ofhe zeros of the zeta function. Based on this analogy and pearls minedom Odlyzko's numerical evidence, Sir Michael Berryproposes that there

    xists a classical dynamical system, asymmetric with respect to time

    eversal, the lengths of whose periodic orbits correspond to the rationalrimes, and whose quantum-mechanical analog has a Hamiltonian witheros equal to the imaginary parts of the nontrivial zeros of the zetaunction. The search for such a dynamical system is one approach toroving the Riemann hypothesis." (Daniel Bump)

    When this conjecture was formulated about 80 years ago, it waspparently no more than an inspired guess. Neither Hilbert nor Plyapecified what operator or even what space would be involved in thisorrespondence. Today, however, that guess is increasingly regarded asonderfuly inspired, and many researchers feel that the most promisingpproach to proving the Riemann Hypothesis is through proving the

    ilbert-Plya conjecture. Their confidence is bolstered by severalevelopments subsequent to Hilbert's and Plya's formulation of theironjecture. There are very suggestive analogies with Selberg zeta

    unctions. There is also the extensive research stimulated by Hugh

    ontgomery's work on the pair-correlation conjecture for zeros of theeta function. Montgomery's results led to the conjecture that zeta zeros

    ehave asynptotically like eigenvalues of large random matrices from the

    UE ensemble that has been studied extensively by mathematical

    hysicists...Although this conjecture is very speculative, the empirical

    vidence is overwhelmingly in its favor."

    .M. Odlyzko from "The 1022-nd zero of the Riemann zeta function".

    We have all this evidence that the Riemann zeros are vibrations but we

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://www.maths.ex.ac.uk/~mwatkins/zeta/odlyzko1.pdfhttp://www.phy.bris.ac.uk/staff/berry_mv.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry.htmhttp://math.stanford.edu/~bump/http://www.maths.ex.ac.uk/~mwatkins/zeta/physics4.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/physics4.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/odlyzko1.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/odlyzko1.pdfhttp://www.dtc.umn.edu/~odlyzko/http://www.maths.ex.ac.uk/~mwatkins/zeta/odlyzko2.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/odlyzko2.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/odlyzko2.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/odlyzko2.pdfhttp://www.dtc.umn.edu/~odlyzko/http://www.maths.ex.ac.uk/~mwatkins/zeta/odlyzko1.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/odlyzko1.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/physics4.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/physics4.htmhttp://math.stanford.edu/~bump/http://www.maths.ex.ac.uk/~mwatkins/zeta/berry.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry.htmhttp://www.phy.bris.ac.uk/staff/berry_mv.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/odlyzko1.pdf
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    . du Sautoy, The Music of the Primes (Fourth Estate, 2003) p.280

    ndrew Odlyko's correspondence with George Plya concerning the

    rigins of the 'Hilbert-Plya conjecture'.

    . du Sautoy describes Jon Keating's discovery in Riemann's notes of

    ossible evidence that he may have considered this approach to hisypothesis long before Hilbert or Plya.

    onference: Arithmetic Quantum Chaos, 23-24 January 2004,

    partement de Mathmatiques, Universit Montpellier, FranceArithmetic Quantum Chaos" is a research area at the cross-roads offferential geometry, ergodic theory, harmonic analysis, mathematicalhysics, and number theory. This session of the MAT Seminar will focusn important recent progress in this area and will consist of two series oftroductory lectures given by experts in the field, with the goal of

    howing that quantum chaos hides a deep harmony at its core.

    he first series of lectures will focus on several aspects of the spectrum

    f Riemann surfaces - on the one hand, the existence and the reparitionf eigenvalues of the laplacian operator, and on the other hand, theroperties of its eigenfunctions (behavior with respect to auasiconformal deformation, properties of equirepartition when thegenvalue goes to infinity, ...). The main focus will be on the case of

    urfaces of "arithmetic" type for which ergodic methods, as well asmethods coming from the theory of automorphic forms and analyticumber theory, have been able to make spectacular progress and torove (at least in the arithmetic case) several of the main conjecturesom quantum chaos theory.

    he second series of lectures will be devoted to random matrices.ntroduced by E. P. Wigner as a way of modelling the resonances of aneavy (atomic) nucleus, this theory has - thanks to the works ofontgomery and more recently Katz/Sarnak - found applications in the

    nderstanding of the zeros of L-functions.

    he most important of these, of course, is the Riemann zeta function. Buthe model becomes especially significant when we consider generalamilies of L-functions of automorphic forms. We then get a coherent

    caffold of conjectures on the structure of the zeros, as well as specialalues, of L-functions. Many of these conjectures have been confirmed

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://www.musicoftheprimes.com/http://www.dtc.umn.edu/~odlyzko/polya/http://www.maths.ex.ac.uk/~mwatkins/zeta/dusautoy-keating.htmhttp://www.math.univ-montp2.fr/~pev/MAT2004/mpl04-en.htmlhttp://www.math.univ-montp2.fr/~pev/MAT2004/mpl04-en.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/dusautoy-keating.htmhttp://www.dtc.umn.edu/~odlyzko/polya/http://www.musicoftheprimes.com/
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    popular/general introductionsMichael Berry's research

    partially successful attempts to produce the required

    operator/dynamics

    other related material

    quantum mechanics and number theory (other contexts)

    quantum chaos links (workgroups, introductory articles, etc.)

    random matrix theory and the Riemann zeta function

    popular/general introductions

    . Cipra, "A Prime Case of Chaos" (An excellent introduction to Berry's

    onjecture, etc. for general readership.)

    . Klarreich, "Prime Time", New Scientist, 11/11/00 (another popular

    xposition)

    The Mark of Zeta" - introductory essay on the Riemann Hypothesis and

    iemann's zeta function (I. Peterson)

    The Return of Zeta" - sequel article by I. Peterson on links between the

    iemann Hypothesis, random matrix theory and quantum chaos

    K ti "Ph i d th Q f M th ti " Ph i W ld A il

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://www.maths.ex.ac.uk/~mwatkins/zeta/qm-general.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/qcresources.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.ams.org/new-in-math/cover/prime-chaos.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/ns111100.htmhttp://www.maa.org/mathland/mathtrek_6_21_99.htmlhttp://www.maa.org/mathland/mathtrek_6_28_99.htmlhttp://www.maa.org/mathland/mathtrek_6_28_99.htmlhttp://www.maa.org/mathland/mathtrek_6_21_99.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/ns111100.htmhttp://www.ams.org/new-in-math/cover/prime-chaos.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/qcresources.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/qm-general.htm
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    ..number theory, once considered by mathematicians to be a field witho application to the other sciences, is now proving to be of considerablese to physicists, both as a working tool and as a guide to possiblerections of future research. It may even be the case that physics canelp in the solution of some of the important problems in this, one of theurest areas of pure mathematics."

    . Cipra, "Prime formula weds number theory and quantum mechanics",cience274 (1996) 2014.

    W. Blum, "Wird dank der Quantenphysik die Riemannsche Vermutung

    ndlich bewiesen?"(an article in German from the newspaperDie Zeiton

    he connection between quantum chaos and the zeros of the Riemanneta function)

    WWN notes - "Physics and the zeros of the zeta-function" (part of a

    ork-in-progress)

    . Patson, "Review of quantum chaology and structural complexity

    pproaches to characterising global behaviour with application to primes"

    A more mathematically advanced introduction to these issues.)

    . Kriecherbauer, J. Marklof and A. Soshnikov, "Random matrices anduantum chaos" (brief introductory article, including a description of how

    his applies to the Riemann Hypothesis)

    .-J. Stckmann, Quantum Chaos: An Introduction (Cambridge

    niversity Press, 1999)

    Michael Berry's research

    he following is an excellent survey article on the work ofBerry and

    eating in this area:

    .V. Berry and J.P. Keating "The Riemann zeros and eigenvalue

    symptotics", SIAM Review, 41, No. 2 (1999) 236-266.

    ee also:

    .V. Berry, "Quantum chaology, prime numbers and Riemann's zeta

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://www.zeit.de/2001/03/Wissen/200103_riemann.htmlhttp://www.zeit.de/2001/03/Wissen/200103_riemann.htmlhttp://aimath.org/WWN/http://aimath.org/WWN/rh/articles/html/31a/http://aimath.org/WWN/rh/http://www.maths.ex.ac.uk/~mwatkins/zeta/patson.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/patson.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/RMTandQC.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/RMTandQC.pdfhttp://www.physicstoday.com/pt/vol-54/iss-1/p49b.htmlhttp://www.phy.bris.ac.uk/staff/berry_mv.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating1.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating1.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry2.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry2.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry2.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry2.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating1.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating1.pdfhttp://www.phy.bris.ac.uk/staff/berry_mv.htmlhttp://www.physicstoday.com/pt/vol-54/iss-1/p49b.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/RMTandQC.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/RMTandQC.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/patson.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/patson.pdfhttp://aimath.org/WWN/rh/http://aimath.org/WWN/rh/articles/html/31a/http://aimath.org/WWN/http://www.zeit.de/2001/03/Wissen/200103_riemann.htmlhttp://www.zeit.de/2001/03/Wissen/200103_riemann.html
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    .V. Berry, "Riemann's zeta function: a model for quantum chaos?"

    .V. Berry and J.P. Keating, "H=xp and the Riemann zeros", from

    upersymmetry and Trace Formulae: Chaos and Disorder, ed. Lerner, et.

    l. (Kluwer/Plenum, 1999).

    .V. Berry, "Semiclassical formula for the number variance of the

    iemann zeros", Nonlinearity1 (1988) 399-407..V. Berry and J.P. Keating, "A new asymptotic representation for

    eta(1/2 + it) and quantum spectral invariants", Proceedings of the Royal

    ociety fo London A437 151-173.

    .V. Berry, "Quantum Chaology" (The Bakerian Lecture, 1987),

    roceedings of the Royal Society of London A 413 (1987) 183-198.

    ichael Berry's list of publications (most are available for downloading asDF files)

    . du Sautoy's account of how Berry came to be involved with the

    iemann Hypothesis

    partially-successful attempts to produce the requiredoperator/dynamics

    everal partially succesful attempts have been made by otheresearchers to produce the required Hermitian operator, and thereby, viaerry's scheme, prove the Riemann Hypothesis.

    .C. Rosu, "Quantum Hamiltonians and prime numbers", Modern

    hysics Letters A 18 (2003)

    abstract:] "A short review of Schrdinger hamiltonians for which thepectral problem has been related in the literature to the distribution ofhe prime numbers is presented here. We notice a possible connectionetween prime numbers and centrifugal inversions in black holes anduggest that this remarkable link could be directly studied within trappedose-Einstein condensates. In addition, when referring to the factorizingperators of Pitkanen and Castro and collaborators, we perform a

    mathematical extension allowing a more standard supersymmetricpproach."

    his very welcome, thorough review article discusses and compares thei i t l t d k f Bh d i Kh L B K ti A

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://www.maths.ex.ac.uk/~mwatkins/zeta/berry5.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating2.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating2.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating2.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating2.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating2.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/disorder.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry3.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry3.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating3.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating3.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating3.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating3.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating3.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry4.pdfhttp://www.phy.bris.ac.uk/research/theory/Berry/publications.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/dusautoy-berry.htmhttp://xxx.lanl.gov/abs/quant-ph/0304139http://xxx.lanl.gov/abs/quant-ph/0304139http://www.maths.ex.ac.uk/~mwatkins/zeta/dusautoy-berry.htmhttp://www.phy.bris.ac.uk/research/theory/Berry/publications.htmlhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry4.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating3.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating3.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry3.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry3.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/disorder.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry-keating2.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry5.pdf
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    oos-Korepin, Crehan and others.

    . Connes, "Formule de trace en geometrie non commutative etypothese de Riemann", C.R.Sci. Paris, t.323, Serie 1 (Analyse) (1996)231-1235.;

    Abstract) "We reduce the Riemann hypothesis forL-functions on a

    obal field kto the validity (not rigorously justified) of a trace formula forhe action of the idele class group on the noncommutative space quotientf the adeles ofkby the multiplicative group ofk."

    erry and Keating refer to this article in their "H=xp and the Riemanneros", and explain that Connes has devised a Hermitian operator whosegenvalues are the Riemann zeros on the critical line. This is almostthe

    perator Berry seeks in order to prove the Riemann Hypothesis, but

    nfortunately the possibility of zeros off the critical line cannot be ruledut in Connes' approach.

    is operator is the transfer (Perron-Frobenius) operator of a classicalansformation. Such classical operators formally resemble quantumamiltonians, but usually have complicated non-discrete spectra andngular eigenfunctions. Connes gets a discrete spectrum by making theperator act on an abstract space where the primes appearing in theuler product for the Riemann zeta function are built in; the space isonstructed from collections ofp-adic numbers (adeles) and thessociated units (ideles). The proof of the Riemann Hypothesis is thus

    educed to the proof of a certain classical trace formula.

    . Deitmar, "A Plya-Hilbert operator for automorphic L-functions"

    abstract:] "We generalize the first part of A. Connes papermath/9811068) on the zeroes of the Riemann zeta function from a

    umber field kto any simple algebra Moverk. To a given automorphicepresentationpiof the reductive group M* of invertible elements ofMwend a Hilbert space Hpiand an operatorDpi(Plya-Hilbert operator),

    hich is the infinitesimal generator of a canonical flow such that thepectrum ofDpicoincides with the purely imaginary zeroes of the function

    (pi,\rez{2}+z). As a byproduct we get meromorphicity of all automorphic-functions, not only the cuspidal ones.

    . Okubo, "Lorentz-invariant Hamiltonian and Riemann Hypothesis"

    We have given some arguments that a two-dimensionalorentz-invariant Hamiltonian may be relevant to the Riemann hypothesisoncerning zero points of the Riemann zeta function. Some eigenfunctionf the Hamiltonian corresponding to infinite-dimensional representation of

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://xxx.lanl.gov/abs/math.NT/9903061http://xxx.lanl.gov/abs/math.NT/9903061http://xxx.lanl.gov/abs/math.NT/9903061http://xxx.lanl.gov/abs/math/9811068http://xxx.lanl.gov/abs/quant-ph/9707036http://xxx.lanl.gov/abs/quant-ph/9707036http://xxx.lanl.gov/abs/math/9811068http://xxx.lanl.gov/abs/math.NT/9903061
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    bsence of trivial representations in the wave function."

    .K. Bhaduri, Avinash Khare, S.M. Reimann, and E.L. Tomusiak, "The

    iemann zeta function and the inverted harmonic oscillator" (outline)

    .K. Bhaduri, Avinash Khare and J. Law, "Phase of the Riemann zetaunction and the inverted harmonic oscillator" article outline

    The Argand diagram is used to display some characteristics of theiemann zeta function...The Argand plots also lead to an analogy with

    he scattering amplitude and an approximate rule for the location of theeros. The smooth phase of the zeta function along the line of the zerosrelated to the quantum density of states of an inverted oscillator."

    The loop structure of the zeta function ...with some near-circular shapes,reminiscent of the Argand plots for the scattering amplitudes offferent partial waves in the analysis of resonances, for example, inon-nucleon scattering."

    V. Armitage, "The Riemann Hypothesis and the Hamiltonian of auantum mechanical system", from Number Theory and Dynamicalystems, eds. M.M. Dodson and J.A.G. Vickers (LMS Lecture Notes,eries 134, Cambridge University Press), 153-172.

    The basic theme of this lecture is an approach to the Riemannypothesis in terms of diffusion processes, which has occupied theuthor's attention for twelve years and which, if correct, has someantalisingly appealing features culminating in a plausible conjecture that

    mplies the truth of that most celebrated of hypotheses. The connectionith diffusion processes suggests that a change of variables (thetroduction of imaginary time) might yield a connection with quantum

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://www.maths.ex.ac.uk/~mwatkins/zeta/harmonic.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/harmonic.htmhttp://xxx.lanl.gov/abs/chao-dyn/9406006http://www.maths.ex.ac.uk/~mwatkins/zeta/harmonic2.htmmailto:[email protected]:[email protected]://www.maths.ex.ac.uk/~mwatkins/zeta/harmonic2.htmhttp://xxx.lanl.gov/abs/chao-dyn/9406006http://www.maths.ex.ac.uk/~mwatkins/zeta/harmonic.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/harmonic.htm
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    erry relating the zeros of the Riemann zeta-function to the Hamiltonian

    f some quantum mechanical system, which in turn makes preciseilbert's original suggestion that the zeros are eigenvalues of someperator and the Riemann Hypothesis is true because that operator isermitian. We shall offer possible candidates for Hilbert's operator anderry's Hamiltonian, but we do not claim satisfactorily to have settled

    hose questions, let alone to have proved the Riemann Hypothesis"

    ua Wu and D.W.L. Sprung, "Riemann zeta and a fractal potential",

    hysical Review E48 (1993) 2595.

    The nontrivial Riemann zeros are reproduced using a one-dimensionalcal-potential model. A close look at the potential suggests that it has aactal structure of dimension d= 1.5."

    . Chadan and M. Musette, "On an interesting singular potential", C.R.cad. Sci. Paris316 II, 1 (1993)

    commentary by H. Rosu:] "Chadan and Musette proposed [a] rather

    omplicated singular potential in a closed interval [0,R] and Dirichletoundary condition at both ends. They gave arguments that the spectrum the coupling constant g= 1/4 + t2/4 (tis the imaginary part on the

    ritical line), which is real and discrete, with gn > 1/4 coincides

    pproximately with the nontrivial Riemann zeros when R=exp(-4*pi/3).

    We note that this is a so-called Sturmian quantum problem, i.e., auantization problem in the coupling constant of the potential. A veryetailed analysis of this singular hamiltonian and a generalization thereofom the point of view of inverse scattering and s-wave Jost functions haseen performed in the important work of Khuri."

    other related material

    here are a number of other approaches:

    . Bump, Kwok-Kwong Choi, P. Kurlberg, and J. Vaaler, "A Local

    iemann Hypothesis, I", Mathematische Zeitschrift233 (1) (2000), 1-18.

    A subscription to Mathematische Zeitschriftis required if you wish toownload this.)

    This paper describes] how local Tate integrals formed withgenfunctions of the quantum mechanical harmonic oscillator, and its

    -adic analogs, have their zeros on the line Re(s) = 1/2.his...incorporates new material on the harmonic oscillator in n

    i M lli t f f th L f ti d i it

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://www.maths.ex.ac.uk/~mwatkins/zeta/berry.htmhttp://www.maths.ex.ac.uk/~mwatkins/zeta/wu_sprung93.pdfhttp://xxx.lanl.gov/abs/quant-ph/0304139http://xxx.lanl.gov/abs/hep-th/0111067http://link.springer.de/link/service/journals/00209/bibs/0233001/02330001.htmhttp://link.springer.de/link/service/journals/00209/bibs/0233001/02330001.htmhttp://link.springer.de/link/service/journals/00209/bibs/0233001/02330001.htmhttp://link.springer.de/link/service/journals/00209/bibs/0233001/02330001.htmhttp://xxx.lanl.gov/abs/hep-th/0111067http://xxx.lanl.gov/abs/quant-ph/0304139http://www.maths.ex.ac.uk/~mwatkins/zeta/wu_sprung93.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/berry.htm
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    . Aneva, "Symmetry of the Riemann operator", Physics Letters B450

    999) 388-396.

    Chaos quantization conditions, which relate the eigenvalues of aermitian operator (the Riemann operator) with the non-trivial zeros of

    he Riemann zeta function are considered, and their geometrical

    terpretation is discussed"

    Sakhr, R.K. Bhaduri and B.P. van Zyl, "Zeta function zeros, powers of

    rimes, and quantum chaos"Phys. Rev. E 68 (2003) 026105

    abstract:] "We present a numerical study of Riemann's formula for thescillating part of the density of the primes and their powers. The formulacomprised of an infinite series of oscillatory terms, one for each zero of

    he zeta function on the critical line and was derived by Riemann in his

    aper on primes assuming the Riemann hypothesis. We show that highesolution spectral lines can be generated by the truncated series at allowers of primes and demonstrate explicitly that teh relative linetensitites are correct. We then derive a Gaussian sum rule foriemann's formula. This is used to analyze the numerical convergence of

    he truncated series The connections to quantum chaos andemiclassical physics are discussed."

    . Crehan, "Chaotic spectra of classically integrable systems"

    We prove that any spectral sequence obeying a certain growth law is theuantum spectrum of an equivalence class of classically integrableon-linear oscillators. This implies that exceptions to the Berry-Tabor ruleor the distribution of quantum energy gaps of clasically integrableystems, are far more numerous than previously believed. In particulare show that for each finite dimension k, there are an infinite number ofassically integrable k-dimensional non-linear oscillators whose quantumpectrum reproduces the imaginary part of zeros on the critical line of the

    iemann zeta function."

    -F. Burnol, "A lower bound in an approximation problem involving the

    eros of the Riemann zeta function"

    Abstract:] "We slightly improve the lower bound of Baez-Duarte,alazard, Landreau and Saias in the Nyman-Beurling formulation of theiemann Hypothesis as an approximation problem. We construct Hilbertpace vectors which could prove useful in the context of the the so called

    Hilbert-Plya idea'."

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://www.maths.ex.ac.uk/~mwatkins/zeta/aneva.pdfhttp://ojps.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PLEEE8000068000002026206000001&idtype=cvips&gifs=Yeshttp://ojps.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PLEEE8000068000002026206000001&idtype=cvips&gifs=Yeshttp://xxx.lanl.gov/abs/chao-dyn/9506014http://www-agat.univ-lille1.fr/~burnol/http://weblib.cern.ch/abstract?math.NT/0103058http://weblib.cern.ch/abstract?math.NT/0103058http://weblib.cern.ch/abstract?math.NT/0103058http://weblib.cern.ch/abstract?math.NT/0103058http://www-agat.univ-lille1.fr/~burnol/http://xxx.lanl.gov/abs/chao-dyn/9506014http://ojps.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PLEEE8000068000002026206000001&idtype=cvips&gifs=Yeshttp://ojps.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PLEEE8000068000002026206000001&idtype=cvips&gifs=Yeshttp://www.maths.ex.ac.uk/~mwatkins/zeta/aneva.pdf
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    nd Chaotic Dynamics6 (2001) 205-210.

    abstract:] "The properties of a fictitious, fermionic, many-body systemased on the complex zeros of the Riemann zeta function are studied.he imaginary part of the zeros are interpreted as mean-fieldngle-particle energies, and one fills them up to a Fermi energy EF. The

    stribution of the total energy is shown to be non-Gaussian, asymmetric,nd independent ofEFin the limit EF-> infinity. The moments of the limit

    stribution are computed analytically. The autocorrelation function, thenite energy corrections, and a comparison with random matrix theory

    re also discussed."

    . Leboeuf and A.G. Monastra, "Quantum thermodynamic fluctuations of

    chaotic Fermi-gas model"

    abstract:] "We investigate the thermodynamics of a Fermi gas whosengle-particle energy levels are given by the complex zeros of the

    iemann zeta function. This is a model for a gas, and in particular for antomic nucleus, with an underlying fully chaotic classical dynamics. Therobability distributions of the quantum fluctuations of the grand potentialnd entropy of the gas are computed as a function of temperature andompared, with good agreement, with general predictions obtained fromandom matrix theory and periodic orbit theory (based on prime

    umbers). In each case the universal and non-universal regimes areentified."

    . Castro, A. Granik, and J. Mahecha, "On SUSY-QM, fractal strings andteps towards a proof of the Riemann hypothesis"

    abstract) "The steps towards a proof of Riemann's conjecture usingpectral analysis are rigorously provided. We prove that the only zeroesf the Rieamnn zeta-function are of the form s = 1/2 + i lambdan. A

    upersymmetric quantum mechanical model is proposed as anternative way to prove the Riemann conjecture, inspired in theilbert-Plya proposal; it uses an inverse scattering approach associated

    ith a system ofp-adic harmonic oscillators. An interpretation of theiemann's fundamental relation Z(s) = Z(1 - s) as a duality relation, fromne fractal string L to another dual fractal string L' is proposed."

    . Odlyzko "Primes, quantum chaos, and computers"

    . Katz and P. Sarnak, "Zeroes of zeta functions and symmetry", Bulletin

    f the AMS, 36 (1999)

    . Sarnak, "Arithmetic quantum chaos", Israeli Mathematical Conference

    di 8 (1995) 183

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://xxx.lanl.gov/abs/nucl-th/0302083http://xxx.lanl.gov/abs/nucl-th/0302083http://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://xxx.lanl.gov/abs/hep-th/0107266http://xxx.lanl.gov/abs/hep-th/0107266http://www.maths.ex.ac.uk/~mwatkins/zeta/odlyzko1.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/katzsarnak.pdfmailto:[email protected]:[email protected]://www.maths.ex.ac.uk/~mwatkins/zeta/katzsarnak.pdfhttp://www.maths.ex.ac.uk/~mwatkins/zeta/odlyzko1.pdfhttp://xxx.lanl.gov/abs/hep-th/0107266http://xxx.lanl.gov/abs/hep-th/0107266http://www.maths.ex.ac.uk/~mwatkins/zeta/random.htmhttp://xxx.lanl.gov/abs/nucl-th/0302083http://xxx.lanl.gov/abs/nucl-th/0302083http://www.maths.ex.ac.uk/~mwatkins/zeta/random.htm
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    iemann zeta function demonstrate the power of the technique. Themethod does not depend on the existence of a symbolic code and mighte a tool for a semiclassical quantization of systems with nonhyperbolicr mixed regular-chaotic dynamics as well."

    Main, V.A. Mandelshtam, G. Wunner and H.S. Taylor, "Riemann zeros

    nd periodic orbit quantization by harmonic inversion"

    abstract:] "In formal analogy with Gutzwiller's semiclassical traceormula, the density of zeros of the Riemann zeta function zeta(z=1/2-iw)an be written as a non-convergent series rho(w)=-pi^{-1} sum_pum_{m=1}^infty ln(p)p^{-m/2} cos(wm ln(p)) with p running over therime numbers. We obtain zeros and poles of the zeta function byarmonic inversion of the time signal which is a Fourier transform ofho(w). More than 2500 zeros have been calculated to about 12 digitrecision as eigenvalues of small matrices using the method ofter-diagonalization. Due to formal analogy of the zeta function with

    utzwiller's periodic orbit trace formula, the method can be applied to thetter to accurately calculate individual semiclassical eigenenergies andesonance poles for classically chaotic systems. The periodic orbituantization is demonstrated on the three disk scattering system as ahysical example."

    Main, P.A. Dando, Dz. Belkic and H S Taylor, "Decimation and

    armonic inversion of periodic orbit signals", J. Phys.A: Math. Gen. 33

    2000) 1247-1263.

    excerpts:] "Introduction. The semiclassical quantization of systems withn underlying chaotic classical dynamics is a nontrivial problem due tohe fact that Gutzwiller's trace formula [1, 2] does not usually converge inhose regions where the eigenenergies or resonances are located.arious techniques have been developed to circumvent the convergenceroblem of periodic orbit theory. Examples are the cycle expansionechnique [3], the Riemann-Siegel-type formula and pseudo-orbitxpansions [4], surface of section techniques [5], and a quantization ruleased on a semiclassical approximation to the spectral staircase [6].

    hese techniques have proven to be very efficient for systems withpecial properties, e.g., the cycle expansion for hyperbolic systems withn existing symbolic dynamics, while the other methods mentioned haveeen used for the calculation of bound spectra.

    ...

    n section 5 we present and compare results for the three-disc scatteringystem as a physical example and the zeros of the Riemann zetaunction as a mathematical model for periodic orbit quantization. Some

    oncluding remarks are given in section 6."more papers by J. Main, et. al.

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://weblib.cern.ch/abstract?chao-dyn/9709009http://weblib.cern.ch/abstract?chao-dyn/9709009http://www.tp1.ruhr-uni-bochum.de/people/main/JPA_preprint.pshttp://www.tp1.ruhr-uni-bochum.de/people/main/JPA_preprint.pshttp://www.tp1.ruhr-uni-bochum.de/people/main/papers.htmlhttp://www.tp1.ruhr-uni-bochum.de/people/main/papers.htmlhttp://www.tp1.ruhr-uni-bochum.de/people/main/JPA_preprint.pshttp://www.tp1.ruhr-uni-bochum.de/people/main/JPA_preprint.pshttp://weblib.cern.ch/abstract?chao-dyn/9709009http://weblib.cern.ch/abstract?chao-dyn/9709009
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    . Doron, "Do spectral trace formulae converge?"

    abstract:] "We evaluate the Gutzwiller trace formula for the level densityf classically chaotic systems by considering the level density in aounded energy range and truncating its Fourier integral. This results in amiting procedure which comprises a convergent semiclassicalpproximation to a well defined spectral quantity at each stage. We testhis result on the spectrum of zeros of the Riemann zeta function,

    btaining increasingly good approximations to the level density. Theourier approach also explains the origin of the convergence problemsncountered by the orbit truncation scheme."

    .V. Andreev, O. Agam, B.D. Simons, B.L. Altschuler, "Quantum chaos,reversible classical dynamics, and random matrix theory", Physicaleview Letters76 (1996) 3497

    . Agam, A.V. Andreev, B.L. Altshuler, "Relations between quantum and

    assical spectral determinants (zeta-functions)"

    We demonstrate that beyond the universal regime correlators ofuantum spectral determinants $\Delta(\epsilon)=\det (\epsilon-\hat{H})$f chaotic systems, defined through an averaging over a wide energyterval, are determined by the underlying classical dynamics through the

    pectral determinant $1/Z(z)=\det (z- {\cal L})$, where $e^{-{\cal L}t}$ ishe Perron-Frobenius operator. Application of these results to theiemann zeta function, allows us to conjecture new relations satisfied by

    his function."

    . Levitin and D. Vassiliev, "Spectral asymptotics, renewal theorem, andhe Berry conjecture for a class of fractals", Proceedings of the London

    Mathematical Society(3) 72 (1996) 188-214.

    . Rubinstein, "Evidence for a spectral interpretation of zeros of

    -functions" (Princeton University Ph.D. thesis, 1998)

    ..provides additional theoretical and numerical evidence connecting theeros ofL-functions (generalization of Riemann Zeta) to eigenvalues ofperators from the classical compact groups (unitary, orthogonal, unitaryymplectic)."

    atti Pitknen, Quantum TGD and how to prove Riemann hypothesis

    3/2/2001)

    During last month further ideas about Riemann hypothesis havemerged and have led to further sharpening of Riemann hypothesis ando p adic particle ph sicist's artic lation for hat it is to be ero of

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://weblib.cern.ch/abstract?chao-dyn/9502025http://xxx.lanl.gov/abs/cond-mat/9602131http://xxx.lanl.gov/abs/cond-mat/9602131mailto:[email protected]://www.math.utexas.edu/~miker/thesis/thesis.htmlhttp://www.math.utexas.edu/~miker/thesis/thesis.htmlmailto:[email protected]://www.physics.helsinki.fi/~matpitka/tgd.html#riemahttp://www.physics.helsinki.fi/~matpitka/tgd.html#riemahttp://www.physics.helsinki.fi/~matpitka/tgd.html#riemahttp://www.physics.helsinki.fi/~matpitka/tgd.html#riemamailto:[email protected]://www.math.utexas.edu/~miker/thesis/thesis.htmlhttp://www.math.utexas.edu/~miker/thesis/thesis.htmlmailto:[email protected]://xxx.lanl.gov/abs/cond-mat/9602131http://xxx.lanl.gov/abs/cond-mat/9602131http://weblib.cern.ch/abstract?chao-dyn/9502025
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    uperconformal invariance of the physical system involved. One canerify Hilbert-Plya hypothesis on basis of the physical picture obtained.his means an explicit construction of the differential operator having the

    moduli squared of the zeros of Riemann Zeta as eigenvalues. Thisperator is product of two operators which are Hermitian conjugates ofach other and have zeros of Riemann Zeta as their eigenvalues. Theacts thatxcorresponds to the real part of conformal weight in this modelnd that one hasx= n/2 for the operators appearing in theepresentations of Super Virasoro, suggest thatx= n/2 is indeed the onlyossible value ofxfor the zeros of Riemann zeta both in real andp-adicontext. Hence Riemann hypothesis would indeed reduce touperconformal invariance."

    . Pitknen, "A further step in the proof of Riemann Hypothesis"

    rchive tutorial mystery new search home contact

    Quantum mechanics: The nontrivial zeros of the Riemann zeta function interpreted as a spectrum of energy levels

    http://www.maths.ex.ac.uk/~mwatkins/zeta/pitkanen2.pdfmailto:[email protected]:[email protected]://www.maths.ex.ac.uk/~mwatkins/zeta/pitkanen2.pdf
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    eta-Function, (Cambridge University Press, 1988)

    .A. Karatsuba, S.M. Voronin, N. Koblitz, The Riemann Zeta-function (de

    ruyter, 1992)

    . Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd edition -

    evised by D. Heath-Brown (Oxford University Press, 1986).

    a poem

    rchive tutorial mystery new search home contact

    Riemann's Zeta Function

    http://www.amazon.com/exec/obidos/ASIN/0521499054/o/qid=981751612/sr=8-1/ref=aps_sr_b_1_1/107-1055617-9006940http://www.amazon.co.uk/exec/obidos/ASIN/3110131706/qid%3D980444338/sr%3D1-14/202-4210080-1902207http://www.amazon.com/exec/obidos/ASIN/0198533691/o/qid=981751980/sr=8-1/ref=aps_sr_b_1_1/107-1055617-9006940http://www.maths.ex.ac.uk/~mwatkins/zeta/zetapoem.htmmailto:[email protected]:[email protected]://www.maths.ex.ac.uk/~mwatkins/zeta/zetapoem.htmhttp://www.amazon.com/exec/obidos/ASIN/0198533691/o/qid=981751980/sr=8-1/ref=aps_sr_b_1_1/107-1055617-9006940http://www.amazon.co.uk/exec/obidos/ASIN/3110131706/qid%3D980444338/sr%3D1-14/202-4210080-1902207http://www.amazon.com/exec/obidos/ASIN/0521499054/o/qid=981751612/sr=8-1/ref=aps_sr_b_1_1/107-1055617-9006940
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    The Riemann Hypothesis] is probably the most basic problem inmathematics, in the sense that it is the intertwining of addition andmultiplication. It's a gaping hole in our understanding..."

    A. Connes, quoted in Dr. Riemann's Zeros (Atlantic, 2002), p.208

    asic introduction to the Riemann Hypothesis (C. Caldwell)

    ric Weisstein's notes on the Riemann Hypothesis

    n-depth examination of issues surrounding the Riemann Hypothesis (D.

    ump)

    ntroduction to the Riemann Hypothesis (K. Spiliopoulos)

    . Pugh's excellent "The Riemann Hypothesis in a Nutshell", including a

    (t) plotting applet

    Brian Conrey, "The Riemann Hypothesis", Notices of the AMS (March

    003) - a very nice, comprehensive introduction to the RH

    Perry's introductory notes on the Riemann Hypothesis

    WWN notes on the Riemann Hypothesis (part of a work-in-progress)

    ritical Strip Explorer v0.67, a wonderful applet produced by Raymond

    anzoni for this site - explore the behaviour of the Riemann zeta function and around the critical strip in a highly visual, interactive way. The

    esulting images are quite astonishing!

    iemann's original eight-page paperostScript, English translation other formats

    Riemann wrote only one article on the theory of numbers, published in859. This paper radically redrew the landscape of the subject. The

    pecific approach to the distribution of prime numbers he developed, bothmple and revolutionary, consists of appealing to Cauchy's theory ofolomorphic functions, which at that time was a relatively recentiscovery."

    G. Tenenbaum and M. Mends France, from The Prime Numbers and

    heir Distribution (AMS, 2000)]

    The Riemann Hypothesis and its generalisations" - WWN notes, part of

    work-in-progress, see also the subsections:

    Generalised Riemann Hypothesis

    The Riemann Hypothesis

    http://www.amazon.co.uk/exec/obidos/ASIN/1843541009/ref%3Dbr_ed_top5_1_5/026-0190221-6772420http://www.utm.edu/research/primes/notes/rh.htmlhttp://mathworld.wolfram.com/RiemannHypothesis.