Hayes (2003) the spectrum of riemannium

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Hayes (2003) the spectrum of riemannium

  1. 1. COMPUTING SCIENCE THE SPECTRUM OF RIEMANNIUM Brian Hayes A reprint from American Scientist the magazine of Sigma Xi, the Scientific Research Society Volume 91, Number 4 July–August, 2003 pages 296–300This reprint is provided for personal and noncommercial use. For any other use, please send arequest to Permissions, American Scientist, P.O. Box 13975, Research Triangle Park, NC, 27709, U.S.A.,or by electronic mail to perms@amsci.org. © 2003 Brian Hayes. 2003
  2. 2. COMPUTING SCIENCE THE SPECTRUM OF RIEMANNIUM Brian Hayes T he year: 1972. The scene: Afternoon tea in games of solitaire, one-dimensional gases and Fuld Hall at the Institute for Advanced chaotic quantum systems. Is it all just a cosmic Study. The camera pans around the Com- coincidence, or is there something going on be- mon Room, passing by several Princetonians in hind the scenes? tweeds and corduroys, then zooms in on Hugh Montgomery, boyish Midwestern number theo- The Spectrum of Interstatium rist with sideburns. He has just been introduced How things distribute themselves in space or time to Freeman Dyson, dapper British physicist. or along some more abstract dimension is a ques- Dyson: So tell me, Montgomery, what have you tion that comes up in all the sciences. An as- been up to? tronomer wants to know how galaxies are scat- Montgomery: Well, lately I’ve been looking into tered around the universe; a biologist might study the distribution of the zeros of the Riemann zeta the distribution of genes along a strand of chro- function. matin; a seismologist records the temporal pattern Dyson: Yes? And? of earthquakes; a mathematician ponders the Montgomery: It seems the two-point correlations sprinkling of prime numbers among the integers. go as.... (turning to write on a nearby blackboard): Here I shall consider only discrete, one-dimen- 2 sional distributions, where the positions of items ( 1 – sin(πx) πx ) can be plotted along a line. Figure 1 shows samples of several such distri- Dyson: Extraordinary! Do you realize that’s the butions, some of them mathematically defined pair-correlation function for the eigenvalues of a and others derived from measurements or obser- random Hermitian matrix? It’s also a model of vations. All of the samples have been scaled so the energy levels in a heavy nucleus—say U-238. that exactly 100 levels fit in the space allotted. I present this anecdote in cinematic form be- Thus the mean distance between levels is the cause I expect to see it on the big screen someday, same in all cases, but the patterns are nonetheless now that mathematicians outgun cowboys and se- quite diverse. For example, the earthquake series cret agents at the box office. Besides, the screen- is highly clustered, which surely reflects some play genre gives me license to dramatize and em- geophysical mechanism. The lower-frequency bellish a little. By the time the movie opens at your fluctuations of tree-ring data probably have both local multiplex, the script doctors will have taken biological and climatological causes. And it’s further liberties with the facts. For example, the anyone’s guess how to explain the locations of equation for nuclear energy levels will have be- bridges recorded while driving along a stretch of come the secret formula of the atomic bomb. Interstate highway. Even without Hollywood hyperbole, however, In analyzing patterns of this kind, there is sel- the chance encounter of Montgomery and Dyson dom much hope of predicting the positions of in- was a genuinely dramatic moment. Their con- dividual elements in a series. The aim is statistical versation revealed an unsuspected connection understanding—a description of a typical pattern between areas of mathematics and physics that rather than a specific one. I shall focus on two sta- had seemed remote. Why should the same equa- tistical measures: nearest-neighbor spacings and tion describe both the structure of an atomic nu- the two-point correlation function. cleus and a sequence at the heart of number the- The simplest of all distributions is a periodic ory? And what do random matrices have to do one. Think of a picket fence or the monotonous with either of those realms? In recent years, the ticking of a clock: All the intervals between ele- plot has thickened further, as random matrices ments of the series are exactly the same. The ob- have turned up in other unlikely places, such as vious counterpoint to such a repetitive pattern is a totally random one. And between these ex- Brian Hayes is Senior Writer for American Scientist. Address: tremes of order and disorder there are various 211 Dacian Avenue, Durham, NC 27701; bhayes@amsci.org intermediate possibilities, such as a “jiggled”296 American Scientist, Volume 91
  3. 3. periodic random jiggled erbium eigenvalues zeta zeros primes bridges railroad tree rings quakes bicyclingFigure 1. One-dimensional distributions each consist of 100 levels. From left to right the spectra are: a periodic array of evenly spaced lines; a ran-dom sequence; a periodic array perturbed by a slight random “jiggling” of each level; energy states of the erbium-166 nucleus, all having the samespin and parity quantum numbers; the central 100 eigenvalues of a 300-by-300 random symmetric matrix; positions of zeros of the Riemann zetafunction lying just above the 1022nd zero; 100 consecutive prime numbers beginning with 103,613; locations of the 100 northernmost overpasses andunderpasses along Interstate 85; positions of crossties on a railroad siding; locations of growth rings from 1884 through 1983 in a fir tree on MountSaint Helens, Washington; dates of California earthquakes with a magnitude of 5.0 or greater, 1969 to 2001; lengths of 100 consecutive bike rides. picket fence, where periodic levels have been James Rainwater and their colleagues at Columbia randomly displaced by a small amount. University. The nucleus in question is that of the A graph of nearest-neighbor spacings readily rare-earth element erbium-166. A glance at the distinguishes among the periodic, random and spectrum reveals no obvious patterns; neverthe- jiggled patterns (see Figure 2). For the periodic less, the texture is quite different from that of a distribution, the graph is a single point: All the purely random distribution. In particular, the er- spacings are the same. The nearest-neighbor bium spectrum has fewer closely spaced levels spectrum of the random distribution is more in- than a random sequence would. It’s as if the nu- teresting: The frequency of any spacing x is pro- clear energy levels come equipped with springs portional to e –x. This negative-exponential law to keep them apart. This phenomenon of “level implies that the smallest spacing between levels repulsion” is characteristic of all heavy nuclei. is the likeliest. The jiggled pattern yields a bell- What kind of mathematical structure could ac- shaped curve, suggesting that the nearest-neigh- count for such a spectrum? This is where those bor intervals have a Gaussian distribution. eigenvalues of random Hermitian matrices enter The pair-correlation function mentioned by the picture. They were proposed for this purpose Montgomery and Dyson captures some of the in the 1950s by the physicist Eugene P. Wigner. As same information as the nearest-neighbor spec- it happens, Wigner was another Princetonian, trum, but it is calculated differently. For each dis- who could therefore make an appearance in our tance x, the correlation function counts how many movie. Let him be the kindly professor who ex- pairs of levels are separated by x, whether or not plains things to a dull student, while the audience those levels are nearest neighbors. The pair-corre- nods knowingly. The dialogue might go like this: lation function for a random distribution is flat, Wigner: Come, we’ll make ourselves a random since all intervals are equally likely. As the distri- Hermitian matrix. We start with a square array, bution becomes more orderly, the pair-correlation like a chessboard, and in each little square we put function develops humps and ripples; for the pe- a random number.... riodic distribution it is a series of sharp spikes. Student: What kind of number? Real? Complex? Wigner: It works with either, but real is easier. Erbium and Eigenvalium Student: And what kind of random? Do we take Among the spectra in Figure 1 is a series of 100 en- them from a uniform distribution, a Gaussian...? ergy levels of an atomic nucleus, measured 30 Wigner: Customarily Gaussian with mean 0 years ago with great finesse by H. I. Liou and and variance 1, but this is not critical. What iswww.americanscientist.org 2003 July–August 297
  4. 4. critical is that the matrix be Hermitian. A Her- When I first heard of the random-matrix con- mitian matrix—it’s named for the French mathe- jecture in nuclear physics, what surprised me matician Charles Hermite—has a special sym- most was not that it might be true but that anyone metry. The main diagonal, running from the would ever have stumbled on it. But Wigner’s upper left to the lower right, acts as a kind of idea was not just a wild guess. In Werner Heisen- mirror, so that all the elements in the upper tri- berg’s formulation of quantum mechanics, the angle are reflected in the lower triangle. internal state of an atom or a nucleus is repre- Student: Then the matrix isn’t really random, sented by a Hermitian matrix whose eigenvalues is it? are the energy levels of the spectrum. If we knew Wigner: If you insist, we’ll call it half-random. the entries in all the columns and rows of this We fill the upper half with random numbers, and matrix, we could calculate the spectrum exactly. then we copy them into the lower half. So now Of course we don’t have that knowledge, but we have our random Hermitian matrix M, and Wigner’s conjecture suggests that the statistics of when we calculate its eigenvalues.... the spectrum are not terribly sensitive to the spe- Student: But how do I do that? cific matrix elements. Thus if we just choose a Wigner: You start up Matlab and you type typical matrix—a large one with elements select- “eig(M)”! ed according to a certain statistical rule—the pre- Eigenvalues go by many names, all of them dictions should be approximately correct. The equally opaque: characteristic values, latent predictions of the model were later worked out roots, the spectrum of a matrix. Definitions, too, more precisely by Dyson and others. are more numerous than helpful. For present purposes it seems best to say that every N-by-N Eulerium and Riemannium matrix is associated with an Nth-degree polyno- So much for nuclear physics; what about number mial equation, and the eigenvalues are the roots theory and the zeta function? of this equation. There are N of them. In general, The most celebrated sequence in number theo- the eigenvalues can be complex numbers, even ry is that of the primes: 2, 3, 5, 7, 11.... The overall when the elements of the matrix are real, but the trend in this series is well known. In the neigh- symmetry of a Hermitian matrix ensures that all borhood of any large integer x, the proportion of the eigenvalues will be real. Hence they can be numbers that are prime is approximately 1/log sorted from smallest to largest and arranged x, which implies that although the primes go on along a line, like energy levels. In this configura- forever, they get sparser as you climb farther out tion they look a lot like the spectrum of a heavy on the number line. Superimposed on this grad- nucleus. Of course the eigenvalues do not match ual thinning of the crop are smaller-scale fluctua- any particular nuclear spectrum level-for-level, tions that are harder to understand in detail. The but statistically the resemblance is strong. sequence of primes looks quite random and er- ratic, and yet it cannot possibly have the same periodic random jiggled nearest-neighbor statistics as a truly random spec- trum. The nearest that two primes can approach each other (except in one anomalous case) is 2. Pairs that have this minimum spacing, such as 29 and 31, are called twin primes. No one knows whether there are infinitely many of them. In addition to directly exploring the primes,0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 mathematicians have taken a roundabout ap- nearest-neighbor spacing proach to understanding their distribution byFigure 2. Nearest-neighbor spacing paints a simple statistical portrait of way of the Riemann zeta function. This function,a distribution. For a periodic series the nearest-neighbor curve has a although named both by and for Bernhard Rie-single nonzero point; for random levels the curve follows an exponen- mann, was first studied in the 18th century bytial law; the “jiggled” series yields a graph that looks Gaussian. Leonhard Euler, who defined it as a sum over all the natural numbers: 1 ζ(s) = Σn n s In other words, take each natural number n from 1 to infinity, raise it to the power s, take the recipro- cal, and add up the entire series. The sum is finite periodic random jiggled whenever s is greater than 1. For example, Euler0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 showed that ζ(2) is equal to π2/6, or about 1.645: inter-pair distance ζ(2) = 12 + 12 + 12 + 12 + ... = π 2Figure 3. Two-point correlation function measures the number of pairs 1 2 3 4 6of levels separated by any given distance. For a random set of levels allintervals are equally likely, whereas the curve for a “jiggled” series has Euler also proved a remarkable identity, equatinga peak at every multiple of the mean nearest-neighbor spacing. the summation formula, with its one term for298 American Scientist, Volume 91
  5. 5. each natural number, to a product formula that 1.0 has one term for each prime. This second defini- tion states: 0.8 –1 ζ(s) = Π( )1– 1s p frequency p 0.6 The recipe in this case is to take each prime p from 2 to infinity, raise it to the power s, then af- 0.4 ter some further arithmetic multiply together the terms for all p. The result is the same as that of 0.2 the summation. This connection between a sum over all integers and a product over all primes 0.0 was a hint that the zeta function might have 0.0 0.5 1.0 1.5 2.0 2.5 3.0 nearest-neighbor spacing something to say about the distribution of primes among the integers, and in fact the two series are Figure 4. Zeros of the Riemann zeta function have near- intimately related. est-neighbor spacings closely matched by the predictions of random-matrix theory. Red dots represent positions of Riemann’s contribution, in 1859, was to extend a billion zeta zeros above the 1023rd such zero; the blue the domain of the zeta function so that it applies line is the predicted spacing. Data for this graph and for not just when s is a real number greater than 1 Figure 5 were supplied by Andrew M. Odlyzko. but when s is any number—positive or negative, real or complex—with the single exception of 1.2 numbers whose real part is equal to 1. Over much of the complex plane the function turns 1.0 out to be wildly oscillatory, crossing from posi- tive to negative values infinitely often. The cross- ing points, where ζ(s) = 0, are called the zeros of 0.8 the zeta function. There is an infinite series of them frequency along the negative real axis, but these are not 0.6 looked upon with great interest. Riemann called attention to a different infinite series of zeros lying 0.4 above and below the real axis in a vertical strip of the complex plane that includes all numbers 0.2 whose real part is between 0 and 1. Riemann cal- culated the locations of the first three of these 0.0 zeros and found that they lie right in the middle of 0.0 0.5 1.0 1.5 2.0 2.5 3.0 the strip, on the “critical line” with real coordinate distance between levels 1 /2. On the basis of this evidence, plus incredible Figure 5. Pair-correlation function for a billion zeta zeros intuition, he conjectured that all the complex near the 1023rd zero also matches the prediction. The blue zeros are on the critical line. This is the Riemann theoretical curve is the function 1 – (sin(π x)/π x)2 dis- hypothesis, widely regarded as the juiciest prize cussed in 1972 by Montgomery and Dyson. Note the evi- plum in all of contemporary mathematics. dence of “level repulsion”: Closely spaced zeros are rare. In the years since Riemann located the first three zeta zeros, quite a few more have been trend is not smooth, and the details of the fluctu- found. A cooperative computing network called ations are all-important. Gaps and clumps in the ZetaGrid, organized by Sebastian Wedeniwski of series of zeta zeros encode information about cor- IBM, has checked 385 billion of them. So far, responding features in the sequence of primes. every one is on the critical line. There’s even a Montgomery’s work on the pair-correlation proof that infinitely many lie on the line, but function of the zeta zeros was a major step to- what’s wanted is a proof that none lie anywhere ward understanding the statistics of the fluctua- else. That goal remains out of reach. tions. And the encounter in Fuld Hall, when it In the meantime, other aspects of the zeta emerged that Montgomery’s correlation formula zeros have come under scrutiny. Assuming that is the same as that for eigenvalues of random all the zeros are indeed on the critical line, what is matrices, ignited further interest. The correlation their distribution along that line? How does their function implies level repulsion among the zeros density vary as a function of the “height,” T, just as it does in the nucleus, producing a defi- above or below the real axis? ciency of closely spaced zeros. As with the primes, the overall trend in the Montgomery’s result is not a theorem; his proof abundance of zeta zeros is known. The trend of it is contingent on the truth of the Riemann Hy- goes the opposite way: Whereas primes get rarer pothesis. But the accuracy of the correlation func- as they get larger, the zeta zeros crowd together tion can be tested by comparing the theoretical with increasing height. The number of zeros in prediction with computed values of zeta zeros. the neighborhood of height T is proportional to Over the past 20 years Andrew M. Odlyzko, now log T, signifying a slow increase. But, again, the at the University of Minnesota, has taken thewww.americanscientist.org 2003 July–August 299
  6. 6. computation of zeta zeros to heroic heights in or- Is that universal operator really out there, wait- der to perform such tests. For this purpose it is ing to be discovered? Will it ever be identified? For not enough to verify that the zeros lie on the crit- the answers to those questions you’ll have to see ical line; the program must accurately measure the movie. I don’t want to give away the ending. the height of each zero along that line, which is a more demanding task. One of Odlyzko’s early Acknowledgments papers was titled “The 1020th zero of the Riemann I owe the title of this column to Oriol Bohigas, who zeta function and 175 million of its neighbors.” mentioned “the spectrum of Riemannium” in a talk at Since then he has gone on to compute even longer the Mathematical Sciences Research Institute in series of consecutive zeros at even greater heights, Berkeley, where I was an embedded journalist in 1999. now reaching the neighborhood of the 10 23rd At that time I also benefitted from discussions with zero. The agreement between predicted and mea- Alan Edelman, Peter Forrester, Andrew Odlyzko, sured correlations is striking, and it gets better Craig Tracy and others. and better with increasing height. Bibliography The Operator of the Universe Berry, M. V., and J. P. Keating. 1999. The Riemann zeros Is it all just a fluke, this apparent link between and eigenvalue asymptotics. SIAM Review 41:236–266. matrix eigenvalues, nuclear physics and zeta Bleher, Pavel, and Alexander Its (eds.) 2001. Random Matrix Models and Their Applications. MSRI Publications, Vol- zeros? It could be, although a universe with such ume 40. Cambridge: Cambridge University Press. chance coincidences in its fabric might be con- Brody, T. A., J. Flores, J. P. French, P. A. Mello, A. Pandey sidered even stranger than one with mysterious and S. S. M. Wong. 1981. Random-matrix physics: Spec- causal connections. trum and strength fluctuations. Reviews of Modern Another possible explanation is that the statisti- Physics 53:385–479. cal distribution seen in these three cases (and in Conrey, J. Brian. 2003. The Riemann hypothesis. Notices of the American Mathematical Society 50:341–353. several others I have not discussed) is simply a Derbyshire, John. 2003. Prime Obsession: Bernhard Riemann very common way for things to organize them- and the Greatest Unsolved Problem in Mathematics. Wash- selves. There is an analogy here with the Gaussian ington: Joseph Henry Press. distribution, which turns up everywhere in nature Diaconis, Persi. 2003. Patterns in eigenvalues: The 70th Josi- because many different processes all lead to it. ah Willard Gibbs lecture. Bulletin of the American Mathe- Whenever multiple independent contributions are matical Society 40:155–178. summed up, the outcome is the familiar bell- Dyson, Freeman J. 1962. Statistical theory of energy levels shaped Gaussian curve: This observation is the of complex systems, I, II and III. Journal of Mathematical Physics 3:140–175. essence of the Central Limit Theorem. Maybe Forrester, P. J., N. C. Snaith and J. J. M. Verbaarschot (eds.) some similar principle makes the eigenvalue dis- 2003. Special issue on random matrix theory. Journal of tribution ubiquitous. Thus for Montgomery and Physics A 36(12):R1–R10. Dyson to come up with the same correlation func- Katz, Nicholas M., and Peter Sarnak. 1999. Zeroes of zeta tion would not be such a long shot after all. functions and symmetry. Bulletin of the American Mathe- Still another view is that the zeros of the zeta matical Society 36:1–26. function really do represent a spectrum—a series Liou, H. I., H. S. Camarda, S. Wynchank, M. Slagowitz, G. Hacken, F. Rahn and J. Rainwater. 1972. Neutron reso- of energy levels just like those of the erbium nu- nance spectroscopy: VIII: The separated isotopes of er- cleus, but generated by the mathematical element bium: evidence for Dyson’s theory concerning level Riemannium. This idea traces back to David spacings. Physical Review C 5:974–1001. Hilbert and George Pólya, who both suggested Mehta, Madan Lal. 1991. Random Matrices. Second edition. (independently) that the zeros of the zeta func- Boston: Academic Press. tion might be the eigenvalues of some unknown Montgomery, Hugh L. 1973. The pair correlation of the zeta function. Proceedings of Symposia in Pure Mathematics Hermitian “operator.” An operator is a mathe- 24:181–193. matical concept that seems on first acquaintance Odlyzko, A. 1987. On the distribution of spacings between rather different from a matrix—it is a function zeros of the zeta function. Mathematics of Computation that applies to functions—but operators too have 48:273–308. eigenvalues, and a Hermitian operator has sym- Odlyzko, A. M. 2001. The 1022nd zero of the Riemann zeta metries that make all the eigenvalues real num- function. In Dynamical, Spectral, and Arithmetic Zeta Func- tions, M. van Frankenhuysen and M. L. Lapidus, eds., pp. bers, just as in the case of a Hermitian matrix. 139–144. Providence: American Mathematical Society. If the Hilbert-Pólya thesis is correct, then ran- Porter, Charles E. 1965. Statistical Theories of Spectra: Fluctu- dom-matrix methods succeed in number theory ations. New York and London: Academic Press. for essentially the same reason they work in nu- Rockmore, Dan, and J. Laurie Snell. 2002. Chance in the clear physics—because the detailed structure of a primes, parts I, II and III. Chance News, 10.10, 11.02, 11.04. large matrix (or operator) is less important than http://www.dartmouth.edu/~chance/chance_news/ its global symmetries, so that any typical matrix news.html with the right symmetries will produce statisti- Tracy, Craig A., and Harold Widom. 1992. Introduction to random matrices. In Geometric and Quantum Aspects of cally similar results. Behind these approximations Integrable Systems, pp. 103–130. Berlin: Springer Verlag. stands some unique Hermitian operator, which Wigner, Eugene P. 1951. On the statistical distribution of the determines the exact position of all the Riemann widths and spacings of nuclear resonance levels. Pro- zeros and hence the distribution of the primes. ceedings of the Cambridge Philosophical Society 47:790–798.300 American Scientist, Volume 91

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