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Some remarks on the moments of $$ \left| {\varsigma \left( {\tfrac{1} {2} + it} \right)} \right| $$ in short intervals

Acta Mathematica Hungarica
Volume/Issue:
Volume 119: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-007-6233-0
Author: A. Ivić

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}
(t ≍ T, T ɛ ≦ G ≪ T, 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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