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] Montgomery , H. L. , Vaughan , R. C. 2007 Multiplicative Number Theory I. Classical Theory Cambridge University Press . [4] Titchmarsh , E. C. , The Riemann Zeta-Function , 2nd ed

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Abstract  

Some new results on power moments of the integral

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$J_k (t,G) = \frac{1} {{\sqrt {\pi G} }}\int_{ - \infty }^\infty { \left| {\varsigma \left( {\tfrac{1} {2} + it + iu} \right)} \right|^{2k} } e^{ - (u/G)^2 } du$$ \end{document}
(tT, T ɛGT, κ ∈ N) are obtained when κ = 1. These results can be used to derive bounds for moments of
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left| {\varsigma \left( {\tfrac{1} {2} + it} \right)} \right|$$ \end{document}
.

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] Edwards , H. M. 2001 Riemann's Zeta Function Dover New York . [4] Groenevelt , W. , The Wilson function transform

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Abstract  

We study the irrational factor function I(n) introduced by Atanassov and defined by

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$I(n) = \prod\nolimits_{\nu = 1}^k {p_\nu ^{1/\alpha _\nu } }$$ \end{document}
, where
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$n = \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } }$$ \end{document}
is the prime factorization of n. We show that the sequence {G(n)/n}n≧1, where G(n) = Πν=1 n I(ν)1/n, is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n).

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Bagchi, B. , The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series , PhD Thesis, Calcutta, Indian Statistical Institute, 1981

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. [7] Titchmarsh , E. C. 1986 The Theory of the Riemann Zeta-Function 2 Oxford University Press Oxford revised by D

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. 1985 The Riemann Zeta-Function John Wiley & Sons New York . [6] Iwaniec , H. , Kowalski , E. 2004 Analytic Number Theory

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's method of exponential sums 126 IVIĆ, A., The Riemann zeta-function , J. Wiley & Sons, Inc., New York, 1985. MR 87d : 11062

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375 – 382 . [8] Heath-Brown , D. R. 1978 The twelfth power moment of the Riemann zeta-function Q. J. Math. 29

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Exponential sums and the Riemann zeta function V Proc. London Math. Soc. 90 1 – 41 10.1112/S0024611504014959 . [13] Ivić , A

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