p204
What sort of insight could numerical studies of the Riemann zeta function give?See the homepage of A. M. Odlyzko: http://www.dtc.umn.edu/ ∼ odlyzko/ . A. R. Booker, ``Turing and the Riemann Hypothesis,’’ Notice Amer. Math. Soc. 53 , 1208 (2006) is also interesting. [Since Odlyzko is a very powerful researcher a lot of other interesting works can be found at this URL; perhaps it may be kinder to give the URL of the directly relevant page, but it should be a good occasion to look at other papersl. (When the author was an organic synthetic chemist, he was taught to pay attention to the papers before and after the one of direct concern. To do this literally in physics and mathematics is almost impossible, but still it is a good instruction.]
There were a lot of people who did not understand why this question was asked here in the Japanese counterpart (some even thought it was only to show off erudition), so this is not asked in the text in the English version.
The core of a model as a tool of description is to summarize empirical facts, and to help us finding a `good’ way to observe them and insight to understand their essence. In the case of the blackbody radiation already explained, a good empirical fitting was a breakthrough to the true insight. Nevertheless, even though a success of a rather simple empirical formula was quite crucial, the curve itself was not directly connected to the insight.
In the case of the zeta function, a precise empirical distribution of the zeros was obtained numerically by Odlyzko, and the model `explaining’ it (the distribution of the eigenvalues of unitary random matrices noticed by Dyson) strongly supported the conjecture (insight?) due to Hilbert and Polya that the zeros of the Riemann zeta function are eigenvalues of a certain positive definite operator describing this world. This is an example that curve-fitting almost directly gaives us insight rather than modeling efforts give us insight.
See:
The 10^22-nd zero of the Riemann zeta function , A. M. Odlyzko. Dynamical, Spectral, and Arithmetic Zeta Functions, M. van Frankenhuysen and M. L. Lapidus, eds., Amer. Math. Soc., Contemporary Math. series, no. 290, 2001, pp. 139-144 ( http://www.dtc.umn.edu/~odlyzko/doc/zeta.10to22.pdf )
A quotation from this article follows:
The main motivation is to obtain further insights into the Hilbert-Polya conjecture, which predicts that the RH (= Riemann hypothesis) is true because zeros of the zeta function correspond to eigenvalues of a positive operator. When this conjecture was formulated about 80 years ago, it was apparently no more than an inspired guess. Neither Hilbert nor Polya specified what operator or even what space would be involved in this correspondence. Today, however, that guess is increasingly regarded as wonderfully inspired, and many researchers feel that the most promising approach to proving the RH is through proving some form of the Hilbert-Polya conjecture. Their confidence is bolstered by several developments subsequent to Hilbert's and Polya's formulation of their conjecture. There are very suggestive analogies with Selberg zeta functions. There is also the extensive research stimulated by Hugh Montgomery's work on the pair-correlation conjecture for zeros of the zeta function [Mon]. Montgomery's results led to the conjecture that zeta zeros behave asymptotically like eigenvalues of large random matrices from the GUE (= Gaussian Unitary Ensemble) ensemble that has been studied extensively by mathematical physicists. This was the conjecture that motivated the computations of [Od1,Od3,Od4] as well as those described in this note. Although this conjecture is very speculative, the empirical evidence is overwhelmingly in its favor.
P. D. Lax, ``Mathematics and Physics,’’ Bull Am Math Soc 45, 135 (2008) also contains some explanations. You can find a big figure demonstrating that the distribution of the spacings between zeros and the correlation function of zero locations also agree the results obtained from the random matrices.
If asked whether the above observations help finding the positive definite operator governing the Riemann zeros, the author must be quite pessimistic, because the spectrum is quite universal. Furthermore, it is not rare that the spectrum does not change very much even if the operator changes considerably. The following review is highly interesting in this respect:
Circular Law Universality
T Tao and V Vu, From the Littlewood-Offord problem to the circular law: Universality of the spectrum distribution of random matrices
Bull Am Math Soc 46, 377 (2009)
*Wigner's famous semicircular law is quite universal: the distribution of elements need not be Gaussian; mean zero and variance 1 are only required.
*The non-Hermitian variant of this is the Circular Law Conjecture. The difficulty of non-Hermitian matrices is the presence of pseudospectrum. This article outlines the proof of the conjecture only obtained recently.
Furthermore, N. A. BAAS and C. F. SKAU, ``The lord of the numbers, Atle Selberg. On his life and mathematics,'' Bull Am Math Soc 45 , 617 (2008) give the author an impression that Selberg is not so interested in the above idea, although
J P Keating and N C Snaith
Random matrix theory and ζ ( 1/ 2 + iT )
Commun Math Phys 214 , 57 (2000)
This paper studied the characteristic polynomials Z(U, θ) of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. The asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z was compute in the large matrix size limit. The results coincide with a theorem of Selberg for the value distribution of log ζ( 1/ 2 + iT ) in the limit T → ∞.
A related recent paper is :
Yan V. Fyodorov, Ghaith A. Hiary, and Jonathan P. Keating
Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta Function
PRL 108 170601 (2012)
Over the past 40 years, inspired by the pioneering work of Montgomery and Odlyzko, considerable evidence has accumulated for connections between the statistical properties of the Riemann zeta function and those of large random matrices.
The spin-glass-like freezing transition that dominates the low-temperature behavior in the statistical mechanical problems also governs the extreme values taken by the characteristic polynomials of random matrices and the zeta function. [The author has not digested the paper.]
Footnote 34 `the arrogance of attempting to change the world’ Addendum
Notice that this was rather a Zeitgeist of the 19th century and we could smell this behind the development of thermodynamics. Y Yamamoto’s Historical development of ideas of thermal science (2008-9) II p185 says: [after the French Revolution]
Not only the ideology of `Libert¥'{e}, Egalit¥'{e}, Fraternit¥'{e},' but also the dream of increasing productivity by controlling and exploiting Nature must have appealed to the young intellects. The industrialism of Saint-Simon (1760-1825) corroborates this. The key to realize this dream was the steam engine.