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High moments of the Riemann zetafunction
 Duke Math. J
"... In 1918 G. Hardy and J. Littlewood proved an asymptotic estimate for the Second moment of the modulus of the Riemann zetafunction on the segment [1/2,1/2+iT] in the complex plane, as T tends to infinity. In 1926 Ingham proved an asymptotic estimate for the fourth moment. However, since Ingham’s res ..."
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Cited by 30 (4 self)
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In 1918 G. Hardy and J. Littlewood proved an asymptotic estimate for the Second moment of the modulus of the Riemann zetafunction on the segment [1/2,1/2+iT] in the complex plane, as T tends to infinity. In 1926 Ingham proved an asymptotic estimate for the fourth moment. However, since Ingham’s result, nobody has proved an asymptotic formula for any higher moment. Recently J. Conrey and A. Ghosh conjectured a formula for the sixth moment. We develop a new heuristic method to conjecture the asymptotic size of both the sixth and eighth moments. Our estimate for the sixth moment agrees with and strongly supports, in a sense made clear in the paper, the one conjectured by Conrey and Ghosh. Moreover, both our sixth and eighth moment estimates agree with those conjectured recently by J. Keating and N. Snaith based on modeling the zetafunction by characteristic polynomials of random matrices from the Gaussian unitary ensemble. Our method uses a conjectural form of the approximate functional equation for the zetafunction, a conjecture on the behavior of additive divisor sums, and D. Goldston and S. Gonek’s mean value theorem for
Moments of the Riemann zetafunction
 Annals of Mathematics 170
, 2009
"... An important problem in analytic number theory is to gain an understanding of the moments ..."
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Cited by 11 (4 self)
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An important problem in analytic number theory is to gain an understanding of the moments
Lower bounds for moments of Lfunctions
 Proc. Natl. Acad. Sci. USA 102
, 2005
"... Abstract. The moments of central values of families of Lfunctions have recently attracted much attention and, with the work of Keating and Snaith, there are now precise conjectures for their limiting values. We develop a simple method to establish lower bounds of the conjectured order of magnitude ..."
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Cited by 10 (5 self)
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Abstract. The moments of central values of families of Lfunctions have recently attracted much attention and, with the work of Keating and Snaith, there are now precise conjectures for their limiting values. We develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of Lfunctions. As an example we work out the case of the family of all Dirichlet Lfunctions to a prime modulus. Resumé. Les moments des valeurs centrales des familles de fonctions L, ont suscité beaucoup d’intêret recemment et, aprés le travail de Keating et Snaith, il y a maintenant des conjectures prcises sur leur valeur. Nous developpons une technique simple pour établir des bornes inférieures sur l’ordre de grandeur conjecturé de plusieurs familles de fonctions L. Un exemple est de considérer la famille des fonctions L de Dirichlet de conducteur premier. A classical question in the theory of the Riemann zetafunction asks for asymptotics of the moments ∫ T 1 ζ(1 2 + it)2kdt where k is a positive integer. A folklore conjecture states that the 2kth moment should be asymptotic to CkT(log T) k2
On the correlation of shifted values of the Riemann zeta function, arXiv:0910.0664v1 [math.NT
"... Abstract. In 2007, assuming the Riemann Hypothesis (RH), Soundararajan [11] proved that R T 0 ζ(1/2 + it)2k dt ≪k,ǫ T(log T) k2 +ǫ for every k positive real number and every ǫ> 0. In this paper I generalized his methods to find upper bounds for shifted moments. We also obtained their lower bounds ..."
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Cited by 3 (0 self)
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Abstract. In 2007, assuming the Riemann Hypothesis (RH), Soundararajan [11] proved that R T 0 ζ(1/2 + it)2k dt ≪k,ǫ T(log T) k2 +ǫ for every k positive real number and every ǫ> 0. In this paper I generalized his methods to find upper bounds for shifted moments. We also obtained their lower bounds and conjectured asymptotic formulas based on Random matrix model, which is analogous to Keating and Snaith’s work. These upper and lower bounds suggest that the correlation of ζ ( 1 2 + it+iα1)  and ζ ( 1 2 + it+iα2)  transition at α1 −α2  ≈ 1 log T α1 − α2  is much larger than. In particular these distribution appear independent when 1 log T.
© Printed in India Real moments of the restrictive factor
, 2007
"... Abstract. Let λ be a real number such that 0 <λ<1. We establish asymptotic formulas for the weighted real moments ∑ n≤x Rλ (n)(1 − n/x), where R(n) = ∏k −1 ν=1 pαν ν is the Atanassov strong restrictive factor function and n = ∏k ν=1 pαν ν is the prime factorization of n. ..."
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Abstract. Let λ be a real number such that 0 <λ<1. We establish asymptotic formulas for the weighted real moments ∑ n≤x Rλ (n)(1 − n/x), where R(n) = ∏k −1 ν=1 pαν ν is the Atanassov strong restrictive factor function and n = ∏k ν=1 pαν ν is the prime factorization of n.
THREE LECTURES ON THE RIEMANN ZETAFUNCTION
, 2004
"... These lectures were delivered at the “International Conference on Subjects Related to the Clay Problems” held at Chonbuk National University, Chonju, Korea in July, 2002. My aim was to give mathematicians and graduate students unfamiliar with analytic number theory an introduction to the theory of t ..."
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These lectures were delivered at the “International Conference on Subjects Related to the Clay Problems” held at Chonbuk National University, Chonju, Korea in July, 2002. My aim was to give mathematicians and graduate students unfamiliar with analytic number theory an introduction to the theory of the Riemann zeta–function focusing, in particular, on the distribution of its zeros. Professor Y. Yildirin of the University
Riemann zeros and random matrix theory
, 2009
"... In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much re ..."
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In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much recent work concerning other varieties of Lfunctions, this article will concentrate on the zeta function as the simplest example illustrating the role of random matrix theory. 1