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ON A NEW CIRCLE PROBLEM

Part of: Multiplicative number theory

Published online by Cambridge University Press:  08 November 2016

JUN FURUYA ,
MAKOTO MINAMIDE  and
YOSHIO TANIGAWA
JUN FURUYA
Affiliation:
Department of Integrated Human Sciences (Mathematics), Hamamatsu University School of Medicine, Handayama 1-20-1, Hamamatsu, Shizuoka431-3192, Japan email jfuruya@hama-med.ac.jp
MAKOTO MINAMIDE*
Affiliation:
Faculty of Science, Yamaguchi University, Yoshida 1677-1, Yamaguchi 753-8512, Japan email minamide@yamaguchi-u.ac.jp
YOSHIO TANIGAWA
Affiliation:
Graduate School of Mathematics, Nagoya University, Furo-cho, Nagoya 464-8602, Japan email tanigawa@math.nagoya-u.ac.jp
*
✉minamide@yamaguchi-u.ac.jp
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Abstract

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We attempt to discuss a new circle problem. Let $\unicode[STIX]{x1D701}(s)$ denote the Riemann zeta-function $\sum _{n=1}^{\infty }n^{-s}$ ($\text{Re}\,s>1$) and $L(s,\unicode[STIX]{x1D712}_{4})$ the Dirichlet $L$-function $\sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$ ($\text{Re}\,s>1$) with the primitive Dirichlet character mod 4. We shall define an arithmetical function $R_{(1,1)}(n)$ by the coefficient of the Dirichlet series $\unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$$(\text{Re}\,s>1)$. This is an analogue of $r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$. In the circle problem, there are many researches of estimations and related topics on the error term in the asymptotic formula for $\sum _{n\leq x}r(n)$. As a new problem, we deduce a ‘truncated Voronoï formula’ for the error term in the asymptotic formula for $\sum _{n\leq x}R_{(1,1)}(n)$. As a direct application, we show the mean square for the error term in our new problem.

Keywords

the circle problemthe truncated Voronoï formuladerivatives of Riemann zeta- and Dirichlet L-functions

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 103 , Issue 2 , October 2017 , pp. 231 - 249
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

This work is supported by JSPS KAKENHI: 26400030, 15K17512, and 15K04778.

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