Results 1  10
of
157
On the distribution of spacings between zeros of the zeta function
 MATH. COMP
, 1987
"... 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 Cray1 and Cray XMP compute ..."
Abstract

Cited by 86 (9 self)
 Add to MetaCart
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 Cray1 and Cray XMP 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
Mellin transforms and asymptotics: Finite differences and Rice's integrals
, 1995
"... High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combin ..."
Abstract

Cited by 82 (8 self)
 Add to MetaCart
High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries, digital search trees, quadtrees, and distributed leader election.
Dynamical Sources in Information Theory: A General Analysis of Trie Structures
 ALGORITHMICA
, 1999
"... Digital trees, also known as tries, are a general purpose flexible data structure that implements dictionaries built on sets of words. An analysis is given of three major representations of tries in the form of arraytries, list tries, and bsttries ("ternary search tries"). The size and the sear ..."
Abstract

Cited by 50 (7 self)
 Add to MetaCart
Digital trees, also known as tries, are a general purpose flexible data structure that implements dictionaries built on sets of words. An analysis is given of three major representations of tries in the form of arraytries, list tries, and bsttries ("ternary search tries"). The size and the search costs of the corresponding representations are analysed precisely in the average case, while a complete distributional analysis of height of tries is given. The unifying data model used is that of dynamical sources and it encompasses classical models like those of memoryless sources with independent symbols, of finite Markovchains, and of nonuniform densities. The probabilistic behaviour of the main parameters, namely size, path length, or height, appears to be determined by two intrinsic characteristics of the source: the entropy and the probability of letter coincidence. These characteristics are themselves related in a natural way to spectral properties of specific transfer operators of the Ruelle type.
Fast algorithms for multiple evaluations of the Riemann zeta function
 Trans. Amer. Math. Soc
, 1988
"... ABSTRACT. The best previously known algorithm for evaluating the Riemann zeta function, c,(a + it), with a bounded and t large to moderate accuracy (within ±t~c for some c> 0, say) was based on the RiemannSiegel formula and required on the order of fc1/2 operations for each value that was computed. ..."
Abstract

Cited by 47 (6 self)
 Add to MetaCart
ABSTRACT. The best previously known algorithm for evaluating the Riemann zeta function, c,(a + it), with a bounded and t large to moderate accuracy (within ±t~c for some c> 0, say) was based on the RiemannSiegel formula and required on the order of fc1/2 operations for each value that was computed. New algorithms are presented in this paper which enable one to compute any single value of c(cr + it) with a fixed and T < t < T + Tlf2 to within ±t~c in 0(te) operations on numbers of O(logi) bits for any e> 0, for example, provided a precomputation involving 0(T1f2+e) operations and 0(T1f2+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 0(n1+s) operations (as opposed to about n3/2 operations for the previous method), as well as improved algorithms for the computation of various arithmetic functions, such as 7r(i). The new zeta function algorithms use the fast Fourier transform and a new method for the evaluation of certain rational functions. They can also be applied to the evaluation of Lfunctions, Epstein zeta functions, and other Dirichlet series. 1. Introduction. Some
Computational Strategies for the Riemann Zeta Function
 Journal of Computational and Applied Mathematics
, 2000
"... 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 th ..."
Abstract

Cited by 46 (9 self)
 Add to MetaCart
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".
On the characteristic polynomial of a random unitary matrix
 Comm. Math. Phys
, 2001
"... Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic√polynomial Z of a random N × N unitary matrix, as N →∞. First 12 we show that ln Z / ln N, evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex n ..."
Abstract

Cited by 43 (11 self)
 Add to MetaCart
Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic√polynomial Z of a random N × N unitary matrix, as N →∞. First 12 we show that ln Z / ln N, evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function. Next we obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lowerorder terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure. Large deviations results for ln Z/A, evaluated at a finite set of distinct points, can be obtained for √ ln N ≪ A ≪ ln N. For higherorder scalings we obtain large deviations results for ln Z/A evaluated at a single point. There is a phase transition at A = ln N (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy.
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... 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 feat ..."
Abstract

Cited by 42 (5 self)
 Add to MetaCart
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 randommatrix 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 RiemannSiegel 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.
Nonvanishing of quadratic Dirichlet Lfunctions at
 Annals of Math. 152 (2000), 447  488 s = 1
"... The Generalized Riemann Hypothesis (GRH) states that all nontrivial zeros of Dirichlet Lfunctions lie on the line Re(s) = 1 2. Further, it is believed that there are no Qlinear ..."
Abstract

Cited by 35 (6 self)
 Add to MetaCart
The Generalized Riemann Hypothesis (GRH) states that all nontrivial zeros of Dirichlet Lfunctions lie on the line Re(s) = 1 2. Further, it is believed that there are no Qlinear
Evidence for a Spectral Interpretation of the Zeros of LFunctions
, 1998
"... By looking at the average behavior (nlevel density) of the low lying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. This is further supported ..."
Abstract

Cited by 33 (7 self)
 Add to MetaCart
By looking at the average behavior (nlevel density) of the low lying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. This is further supported by numerical experiments for which an efficient algorithm to compute Lfunctions 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 Lfunctions. Zeev Rudnick and Andrew Oldyzko for many disc...
Disproof of the Mertens conjecture
 J. REINE ANGEW. MATH
, 1985
"... The Mertens conjecture states that ⎪ M(x) ⎪ < x 1 ⁄ 2 for all x> 1, where M(x) = Σ μ(n), n ≤ x and μ(n) is the Möbius function. This conjecture has attracted a substantial amount of interest in its almost 100 years of existence because its truth was known to imply the truth of the Riemann hypothesi ..."
Abstract

Cited by 29 (3 self)
 Add to MetaCart
The Mertens conjecture states that ⎪ M(x) ⎪ < x 1 ⁄ 2 for all x> 1, where M(x) = Σ μ(n), n ≤ x and μ(n) is the Möbius function. This conjecture has attracted a substantial amount of interest in its almost 100 years of existence because its truth was known to imply the truth of the Riemann hypothesis. This paper disproves the Mertens conjecture by showing that lim sup M(x) x x → ∞ − 1 ⁄ 2> 1. 06. The disproof relies on extensive computations with the zeros of the zeta function, and does not provide an explicit counterexample.