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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 $\unicode[STIX]{x1D701}^{\prime }(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$

  • Jun Furuya (a1), T. Makoto Minamide (a2) and Yoshio Tanigawa (a3)

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.

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

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