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119
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 ..."
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Cited by 86 (9 self)
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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
Probability laws related to the Jacobi theta and Riemann zeta functions, and the Brownian excursions
 Bulletin (New series) of the American Mathematical Society
"... Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional ..."
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Cited by 57 (11 self)
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Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws. Contents
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. ..."
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Cited by 47 (6 self)
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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 ..."
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Cited by 46 (9 self)
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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".
New results on the Stieltjes constants: Asymptotic and exact evaluation
 J. Math. Anal. Appl
, 2006
"... The Stieltjes constants γk(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about s = 1. We present new asymptotic, summatory, and other exact expressions for these and related constants. Key words and phrases Stieltjes constants, Riemann zeta function, Hurwitz z ..."
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Cited by 10 (6 self)
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The Stieltjes constants γk(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about s = 1. We present new asymptotic, summatory, and other exact expressions for these and related constants. Key words and phrases Stieltjes constants, Riemann zeta function, Hurwitz zeta function, Laurent expansion, integrals of periodic Bernoulli polynomials, functional equation, Kreminski
On the mean square of the zetafunction and the divisor problem, Annales Acad. Scien. Fennicae Mathematica
"... Abstract. Let ∆(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of ζ ( 1 + it). 2 If E ∗ (t) = E(t) − 2π∆ ∗ (t/2π) with ∆ ∗ (x) = −∆(x) + 2∆(2x) − 1 ∆(4x), then we ..."
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Cited by 9 (9 self)
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Abstract. Let ∆(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of ζ ( 1 + it). 2 If E ∗ (t) = E(t) − 2π∆ ∗ (t/2π) with ∆ ∗ (x) = −∆(x) + 2∆(2x) − 1 ∆(4x), then we
ON THE DIVISOR FUNCTION AND THE RIEMANN ZETAFUNCTION IN SHORT INTERVALS
, 2007
"... ... where ∆(x) is the error term in the classical divisor problem, and E(T) is the error term in the mean square formula for ζ ( 1 + it). Upper bounds of the form 2 Oε(T 1+εU2) for the above integrals with biquadrates instead of square are shown to hold for T3/8 ≤ U = U(T) ≪ T1/2. The connection ..."
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Cited by 7 (5 self)
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... where ∆(x) is the error term in the classical divisor problem, and E(T) is the error term in the mean square formula for ζ ( 1 + it). Upper bounds of the form 2 Oε(T 1+εU2) for the above integrals with biquadrates instead of square are shown to hold for T3/8 ≤ U = U(T) ≪ T1/2. The connection between the moments of E(t + U) − E(t) and ζ ( 1 + it)  is also given. Generalizations to some other 2 numbertheoretic error terms are discussed.
On a diophantine problem with two primes and s powers of 2, Acta Arith
, 2010
"... We refine a recent result of Parsell [22] on the values of the form λ1p1+λ2p2+µ12 m1 ·· · + µs2 ms, where p1, p2 are prime numbers, m1,...,ms are positive integers, λ1/λ2 is negative and irrational and λ1/µ1, λ2/µ2 ∈ Q. ..."
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Cited by 7 (2 self)
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We refine a recent result of Parsell [22] on the values of the form λ1p1+λ2p2+µ12 m1 ·· · + µs2 ms, where p1, p2 are prime numbers, m1,...,ms are positive integers, λ1/λ2 is negative and irrational and λ1/µ1, λ2/µ2 ∈ Q.
Digital Sums And DivideAndConquer Recurrences: Fourier Expansions And Absolute Convergence
, 2004
"... We propose means of computing the Fourier expansions of periodic functions appearing in higher moments of the sumofdigits function and in the solutions of some divideandconquer recurrences. The expansions are shown to be absolutely convergent. We also give a new approach to efficiently computing ..."
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Cited by 7 (2 self)
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We propose means of computing the Fourier expansions of periodic functions appearing in higher moments of the sumofdigits function and in the solutions of some divideandconquer recurrences. The expansions are shown to be absolutely convergent. We also give a new approach to efficiently computing numerically the coefficients involved to high precision.
ON THE RIEMANN ZETAFUNCTION AND THE DIVISOR PROBLEM II
, 2004
"... Abstract. Let ∆(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of ζ ( 1 + it). 2 If E ∗ (t) = E(t) − 2π∆ ∗ (t/2π) with ∆ ∗ (x) = −∆(x) + 2∆(2x) − 1 ∆(4x), then we 2 obtain ..."
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Cited by 6 (6 self)
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Abstract. Let ∆(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of ζ ( 1 + it). 2 If E ∗ (t) = E(t) − 2π∆ ∗ (t/2π) with ∆ ∗ (x) = −∆(x) + 2∆(2x) − 1 ∆(4x), then we 2 obtain