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208
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 120 (10 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
Computational strategies for the Riemann zeta function
, 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 67 (11 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”.
Fast algorithms for multiple evaluations of the Riemann zeta function
 TRANS. AMER. MATH. SOC
, 1988
"... 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 a ..."
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Cited by 65 (8 self)
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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.
The distribution and moments of the error term in the Dirichlet divisor problem
 Acta Arith
, 1992
"... This paper will consider results about the distribution and moments of some of the well known error terms in analytic number theory. To focus attention we begin by considering the error term ∆(x) in the Dirichlet divisor problem, which is defined as ..."
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Cited by 50 (0 self)
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This paper will consider results about the distribution and moments of some of the well known error terms in analytic number theory. To focus attention we begin by considering the error term ∆(x) in the Dirichlet divisor problem, which is defined as
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 21 (17 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 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 19 (6 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 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 15 (10 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
A complete Vinogradov 3primes theorem under the Riemann hypothesis
 ERA Am. Math. Soc
, 1997
"... Abstract. We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation. 1. ..."
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Cited by 15 (1 self)
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Abstract. We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation. 1.
Series representations for the Stieltjes constants
, 2009
"... The Stieltjes constants γk(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about s = 1. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives a ..."
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Cited by 11 (9 self)
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The Stieltjes constants γk(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about s = 1. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for derivatives of all orders of the Stieltjes coefficients are given. Many other results are obtained, including instances of an exponentially fast converging series representation for γk = γk(1). Some extensions are briefly described, as well as the relevance to expansions of Dirichlet L functions. Key words and phrases Stieltjes constants, Riemann zeta function, Hurwitz zeta function, Laurent expansion, Stirling numbers of the first kind, Dirichlet L functions, Lerch zeta function 2000 AMS codes
On the higher moments of the error term in the divisor problem
"... Abstract. Let ∆(x) denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas and ..."
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Cited by 11 (8 self)
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Abstract. Let ∆(x) denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas and