A numerical study of the distribution of spacings between zeros of the Riemann zeta function is presented. It is based on values for the first 10 5 zeros and for zeros number 10 12 + 1 to 10 12 + 10 5 that are accurate to within ± 10 − 8, and which were calculated on the Cray-1 and Cray X-MP computers. This study tests the Montgomery pair correlation conjecture as well as some further conjectures that predict that the zeros of the zeta function behave similarly to eigenvalues of random hermitian matrices. Matrices of this type are used in modeling energy levels in physics, and many statistical properties of their eigenvalues are known. The agreement between actual statistics for zeros of the zeta function and conjectured results is generally good, and improves at larger heights. Several initially unexpected phenomena were found in the data and some were explained by

The best currently known algorithm for evaluating the Riemann zeta function, z(s + it ), with s bounded and t large to moderate accuracy (within ± t - c for some c > 0, say) is based on the Riemann-Siegel formula and requires on the order of t 1/2 operations for each value that is computed. New algorithms are presented in this paper which enable one to compute any single value of z(s + it ) with s fixed and T t T + T 1/2 to within ± t - c in O(t e ) operations on numbers of O(log t ) bits for any e > 0, for example, provided a precomputation involving O(T 1/2 + e ) operations and O(T 1/2 + e ) bits of storage is carried out beforehand. These algorithms lead to methods for numerically verifying the Riemann Hypothesis for the first n zeros in what is expected to be O(n 1 + e ) operations (as opposed to about n 3/2 operations for the previous method), as well as improved algorithms for the computation of various arithmetic functions, such as p(x). The new zeta functi...

Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.

We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call "value recycling".

By looking at the average behavior (n-level density) of the low lying zeros of certain families of L-functions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the non-trivial zeros in terms of the classical compact groups. This is further supported by numerical experiments for which an efficient algorithm to compute L-functions was developed and implemented. iii Acknowledgements When Mike Rubinstein woke up one morning he was shocked to discover that he was writing the acknowledgements to his thesis. After two screenplays, a 40000 word manifesto, and many fruitless attempts at making sushi, something resembling a detailed academic work has emerged for which he has people to thank. Peter Sarnak- from Chebyshev's Bias to USp(1). For being a terrific advisor and teacher. For choosing problems suited to my talents and involving me in this great project to understand the zeros of L-functions. Zeev Rudnick and Andrew Oldyzko for many disc...

Abstract. Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli k ≤ 72 and other small moduli. 1.

In recent years there has been a growing interest in connections between the statistical properties of number theoretical L-functions and random matrix theory. We review the history of these connections, some of the major achievements, and a number of applications.

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. The de Bruijn-Newman constant has been investigated extensively because the truth of the Riemann Hypothesis is equivalent to the assertion that 0. On the other hand, C. M. Newman conjectured that 0. This paper improves previous lower bounds by showing that \Gamma5:895 \Delta 10 \Gamma9 ! : This is done with the help of a spectacularly close pair of consecutive zeros of the Riemann zeta function. Key words. Lehmer pairs of zeros, de Bruijn-Newman constant, Riemann Hypothesis. AMS subject classifications. 30D10, 30D15, 65E05. 1. Introduction. It is known (cf. Titchmarsh [9, p. 255]) that the Riemann ¸-function can be expressed in the form ¸ i x 2 j =8 = Z 1 0 \Phi(u) cos(xu)du (x 2 I C); (1.1) where \Phi(u) := 1 X n=1 \Gamma 2ß 2 n 4 e 9u \Gamma 3ßn 2 e 5u \Delta exp \Gamma \Gammaßn 2 e 4u \Delta (0 u ! 1); (1.2) and the Riemann Hypothesis is the statement that all zeros of ¸ are real. If we define H t (x) := Z 1 0 e tu 2 \Phi(u) cos(xu)du...

The Riemann Hypothesis, which specifies the location of zeros ofthe Riemann zeta function, and thus describes the behavior of primes, is one of the most famous unsolved problems inmathematics, and extensive efforts have been made over more than a century to check it numerically for large sets of cases. Recently a new algorithm, invented by the speaker and A. Schönhage, has been implemented, and used to compute over 175 million zeros near zero number 10^20. The new algorithm turned out to be over 5 orders of magnitude faster than older methods. The crucial ingredients in it are a rational function evaluation method similar to the Greengard-Rokhlin gravitational potential evaluation algorithm, the FFT, andband-limited function interpolation. While the only present implementation is on a Cray, the algorithm can easily be parallelized.