On the Asymptotics of the Riemann Zeta Function to all Orders

Details On the Asymptotics of the Riemann Zeta Function to all Orders On the Asymptotics of the Riemann Zeta Function to all Orders

2012-01-12

arxiv.org/abs/1201.2633v1

Mathematics

Number Theory

62 pages

Scientific paper

We present several formulae for the large $t$ asymptotics of the Riemann zeta function $\zeta(s)$, $s=\sigma+i t$, $0\leq \sigma \leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum $\sum_a^b n^{-s}$ for certain ranges of $a$ and $b$. In addition, we present precise estimates relating this sum with the sum $\sum_c^d n^{s-1}$ for certain ranges of $a, b, c, d$. Finally, we derive certain novel integral representations for the basic sum characterising the leading large $t$ asymptotics of $\zeta(s)$.

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