1 results
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
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- Journal:
- Canadian Journal of Mathematics / Volume 71 / Issue 6 / December 2019
- Published online by Cambridge University Press:
- 09 January 2019, pp. 1465-1493
- Print publication:
- December 2019
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- Article
- Export citation
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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.
