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Titchmarsh’s Method for the Approximate Functional Equations for $\unicode[STIX]{x1D701}^{\prime }(s)^{2}$, $\unicode[STIX]{x1D701}(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$, and $\unicode[STIX]{x1D701}^{\prime }(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  09 January 2019

Jun Furuya ,
T. Makoto Minamide  and
Yoshio Tanigawa
Jun Furuya
Affiliation:
Department of Integrated Human Sciences (Mathematics), Hamamatsu University School of Medicine, Handayama 1-20-1, Shizuoka 431-3192, Japan Email: jfuruya@hama-med.ac.jp
T. Makoto Minamide
Affiliation:
Graduate School of Sciences and Technology for Innovation, 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
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Abstract

Let $\unicode[STIX]{x1D701}(s)$ be the Riemann zeta function. In 1929, Hardy and Littlewood proved the approximate functional equation for $\unicode[STIX]{x1D701}^{2}(s)$ with error term $O(x^{1/2-\unicode[STIX]{x1D70E}}((x+y)/|t|)^{1/4}\log |t|)$, where $-1/2<\unicode[STIX]{x1D70E}<3/2,x,y\geqslant 1,xy=(|t|/2\unicode[STIX]{x1D70B})^{2}$. Later, in 1938, Titchmarsh improved the error term by removing the factor $((x+y)/|t|)^{1/4}$. In 1999, Hall showed the approximate functional equations for $\unicode[STIX]{x1D701}^{\prime }(s)^{2},\unicode[STIX]{x1D701}(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$, and $\unicode[STIX]{x1D701}^{\prime }(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ (in the range $0<\unicode[STIX]{x1D70E}<1$) whose error terms contain the factor $((x+y)/|t|)^{1/4}$. In this paper we remove this factor from these three error terms by using the method of Titchmarsh.

Keywords

derivative of the Riemann zeta functionapproximate functional equationexponential sum

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 6 , December 2019 , pp. 1465 - 1493
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

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

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