Power moments of the error term for the approximate functional equation of the Riemann zeta-function

Isao Kiuchi

Abstract: Let $\zeta(s)$ be the Riemann zeta-function, $d(n)$ the number of positive divisors of the integer $n$, and $$ R(s;t/2\pi) =\zeta^2(s) -\sum_{n\le t/2\pi}\!\!\!\strut'\enskip d(n)n^{-s} -\chi^2(s) \sum_{n\le t/2\pi}\!\!\!\strut'\enskip d(n)n^{s-1}, $$ where $$ \chi(s)=2^s\pi^{s-1}\sin(\frac12\pi s)\Gamma(1-s). $$ We obtain the following power moment estimates: $$ \int_1^T |R(\frac12+it;t/2\pi)|^A\,dt \ll \cases T^{1-\frac14A+\vaeepsilon},&0\le A\le4,
1,&A>4.\endcases $$

Classification (MSC2000): 11M06

Full text of the article:

• Compressed PostScript file (42 kilobytes)
• PDF file (120 kilobytes)