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20
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
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 ..."
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Cited by 42 (5 self)
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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.
Higher correlations of divisor sums related to primes, II: Variations of . . .
, 2007
"... We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the ..."
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Cited by 28 (6 self)
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We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet Lfunctions, we obtain an Ω±result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ωresults for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes.
Primes in short intervals
 Commun. Math. Phys
"... Dedicated to Freeman Dyson, with best wishes on the occasion of his eightieth birthday. Abstract. Contrary to what would be predicted on the basis of Cramér’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x + H) − ψ(x), for 0 ≤ x ≤ N, is approxima ..."
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Cited by 9 (3 self)
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Dedicated to Freeman Dyson, with best wishes on the occasion of his eightieth birthday. Abstract. Contrary to what would be predicted on the basis of Cramér’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x + H) − ψ(x), for 0 ≤ x ≤ N, is approximately normal with mean ∼ H and variance ∼ H log N/H, when N δ ≤ H ≤ N 1−δ. Cramér [4] modeled the distribution of prime numbers by independent random variables Xn (for n ≥ 3) that take the value 1 (n is “prime”) with probability 1 / logn and take the value 0 (n is “composite”) with probability 1 − 1 / log n. If pn denotes the n th prime
On the pair correlation of the zeros of the Riemann zetafunction
 Proc. London Math. Soc
"... In 1972 Montgomery [20, 21] introduced a new method for studying the zeros of the Riemann zetafunction. One of his main accomplishments was to determine partially the pair correlation of zeros, and to apply his results to obtain new information on multiplicity of zeros and gaps between zeros. Perha ..."
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Cited by 8 (3 self)
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In 1972 Montgomery [20, 21] introduced a new method for studying the zeros of the Riemann zetafunction. One of his main accomplishments was to determine partially the pair correlation of zeros, and to apply his results to obtain new information on multiplicity of zeros and gaps between zeros. Perhaps more
Zero Spacing Distributions for Differenced LFunctions
, 2006
"... The paper studies the local zero spacings of deformations of the Riemann ξfunction under certain averaging and differencing operations. For real h we consider the entire functions Ah(s): = 1 2 (ξ(s + h) + ξ(s − h)) and Bh(s) = 1 ..."
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Cited by 5 (2 self)
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The paper studies the local zero spacings of deformations of the Riemann ξfunction under certain averaging and differencing operations. For real h we consider the entire functions Ah(s): = 1 2 (ξ(s + h) + ξ(s − h)) and Bh(s) = 1
On the second moment for primes in an arithmetic progression, Acta Arithmetica C.1
, 2001
"... Abstract. Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results were averaged over all progression of a given mod ..."
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Cited by 3 (2 self)
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Abstract. Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results were averaged over all progression of a given modulus. The method uses a short divisor sum approximation for the von Mangoldt function, together with some new results for binary correlations of this divisor sum approximation in arithmetic progressions. 1. Introduction and Statement of
Yıldırım, Primes in short segments of arithmetic progressions
 Canad. J. Math
, 1998
"... ABSTRACT. Consider the variance for the number of primes that are both in the interval [y, y + h] for y 2 [x,2x] and in an arithmetic progression of modulus q. We study the total variance obtained by adding these variances over all the reduced residue classes modulo q. Assuming a strong form of the ..."
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Cited by 2 (2 self)
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ABSTRACT. Consider the variance for the number of primes that are both in the interval [y, y + h] for y 2 [x,2x] and in an arithmetic progression of modulus q. We study the total variance obtained by adding these variances over all the reduced residue classes modulo q. Assuming a strong form of the twin prime conjecture and the Riemann Hypothesis one can obtain an asymptotic formula for the total variance in the range when 1 hÛq x 1Û2 è, for anyèÙ0. We show that one can still obtain some weaker asymptotic results assuming the Generalized Riemann Hypothesis (GRH) in place of the twin prime conjecture. In their simplest form, our results are that on GRH the same asymptotic formula obtained with the twin prime conjecture is true for “almost all ” q in the range 1 hÛq h 1Û4 è, that on averaging over q one obtains an asymptotic formula in the extended range 1 hÛq h 1Û2 è, and that there are lower bounds with the correct order of magnitude for all q in the range 1 hÛq x 1Û3 è.
Equivalence of higher moments of primes in short intervals, preprint
"... In this article, we prove an “equivalence ” between two higher even moments of primes in short intervals under Riemann Hypothesis. We also provide numerical evidence in support of these asymptotic formulas. ..."
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Cited by 2 (2 self)
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In this article, we prove an “equivalence ” between two higher even moments of primes in short intervals under Riemann Hypothesis. We also provide numerical evidence in support of these asymptotic formulas.
Quantum chaos, random matrix theory, and the Riemann ζfunction
, 2010
"... Hilbert and Pólya put forward the idea that the zeros of the Riemann zeta function may have a spectral origin: the values of tn such that 1 2 + itn is a non trivial zero of ζ might be the eigenvalues of a selfadjoint operator. This would imply the Riemann Hypothesis. From the perspective of Physics ..."
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Cited by 2 (0 self)
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Hilbert and Pólya put forward the idea that the zeros of the Riemann zeta function may have a spectral origin: the values of tn such that 1 2 + itn is a non trivial zero of ζ might be the eigenvalues of a selfadjoint operator. This would imply the Riemann Hypothesis. From the perspective of Physics one might go further and consider the possibility that the operator in question corresponds to the quantization of a classical dynamical system. The first significant evidence in support of this spectral interpretation of the Riemann zeros emerged in the 1950’s in the form of the resemblance between the Selberg trace formula, which relates the eigenvalues of the Laplacian and the closed geodesics of a Riemann surface, and the Weil explicit formula in number theory, which relates the Riemann zeros to the primes. More generally, the Weil explicit formula resembles very closely a general class of Trace Formulae, written down by Gutzwiller, that relate quantum energy levels to classical periodic orbits in chaotic Hamiltonian systems. The second