§25.9 Asymptotic Approximations

Notes:: See Titchmarsh (1986b, pp. 79 and 88), Berry (1995).
Keywords:: Riemann zeta function
Permalink:: http://dlmf.nist.gov/25.9

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 $\mathop{\zeta\/}\nolimits\!\left(\sigma+it\right)=\sum _{{1\leq n\leq x}}\frac{1}{n^{s}}+\chi(s)\sum _{{1\leq n\leq y}}\frac{1}{n^{{1-s}}}+\mathop{O\/}\nolimits\!\left(x^{{-\sigma}}\right)+\mathop{O\/}\nolimits\!\left(y^{{\sigma-1}}t^{{\frac{1}{2}-\sigma}}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $n$ : nonnegative integer, $x$ : real variable, $s$ : complex variable, $\sigma$ : fixed and $\chi(s)$ : function
Referenced by:: §25.9
Permalink:: http://dlmf.nist.gov/25.9.E1
Encodings:: TeX, pMML, png

where $s=\sigma+it$ and

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $s$ : complex variable and $\chi(s)$ : function
Permalink:: http://dlmf.nist.gov/25.9.E2
Encodings:: TeX, pMML, png

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 $\mathop{\zeta\/}\nolimits\!\left(\tfrac{1}{2}+it\right)=\sum _{{n=1}}^{m}\frac{1}{n^{{\frac{1}{2}+it}}}+\chi\left(\tfrac{1}{2}+it\right)\sum _{{n=1}}^{m}\frac{1}{n^{{\frac{1}{2}-it}}}+\mathop{O\/}\nolimits\!\left(t^{{-1/4}}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : 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

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