§25.9 Asymptotic Approximations

If $x\geq 1$, $y\geq 1$, $2\pi xy=t$, and $0\leq\sigma\leq 1$, then as $t\to\infty$ with $\sigma$ fixed,

 25.9.1 $\zeta\left(\sigma+it\right)=\sum_{1\leq n\leq x}\frac{1}{n^{s}}+\chi(s)\sum_{1% \leq n\leq y}\frac{1}{n^{1-s}}+O\left(x^{-\sigma}\right)+O\left(y^{\sigma-1}t^% {\frac{1}{2}-\sigma}\right),$

where $s=\sigma+it$ and

 25.9.2 $\chi(s)=\pi^{s-\frac{1}{2}}\Gamma\left(\tfrac{1}{2}-\tfrac{1}{2}s\right)/% \Gamma\left(\tfrac{1}{2}s\right).$

If $\sigma=\frac{1}{2}$, $x=y=\sqrt{t/(2\pi)}$, and $m=\left\lfloor x\right\rfloor$, then (25.9.1) becomes

 25.9.3 $\zeta\left(\tfrac{1}{2}+it\right)=\sum_{n=1}^{m}\frac{1}{n^{\frac{1}{2}+it}}+% \chi\left(\tfrac{1}{2}+it\right)\sum_{n=1}^{m}\frac{1}{n^{\frac{1}{2}-it}}+O% \left(t^{-1/4}\right).$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $m$: nonnegative integer, $n$: nonnegative integer and $\chi(s)$: function Referenced by: §25.10(ii) Permalink: http://dlmf.nist.gov/25.9.E3 Encodings: TeX, pMML, png See also: Annotations for 25.9 and 25

For other asymptotic approximations see Berry and Keating (1992), Paris and Cang (1997); see also Paris and Kaminski (2001, pp. 380–389).