Acta Mathematica Hungarica
Volume/Issue:
Volume 129: Issue 1-2
Explicit representations of the integral containing the error term in the divisor problem
Authors: J. Furuya 1 and Y. Tanigawa 2
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  • 1 Department of Integrated Arts and Science, Okinawa National College of Technology, Nago, Okinawa, 905-2192 Japan
  • 2 Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602 Japan
DOI:
https://doi.org/10.1007/s10474-010-9219-2
Page Count:
24–46
Publication Date:
01 Oct 2010
Online Publication Date:
19 Mar 2010
Article Category:
Research Article
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Abstract  

We shall investigate several properties 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} $$\int_1^\infty {t^{ - \theta } \Delta _k \left( t \right) log^j t dt}$$ \end{document}
with a natural number k, a non-negative integer j and a complex variable θ, where Δk(x) is the error term in the divisor problem of Dirichlet and Piltz. The main purpose of this paper is to apply the “elementary methods” and the “elementary formulas” to derive convergence properties and explicit representations of this integral with respect to θ for k = 2.

Cover Acta Mathematica Hungarica
Acta Mathematica Hungarica
Print ISSN:
0236-5294
Online ISSN:
1588-2632
Keywords:
convergence of integral; divisor problems; Riemann zeta-function; 11N37

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