25 Zeta and Related FunctionsRiemann Zeta Function25.8 Sums25.10 Zeros

§25.9 Asymptotic Approximations

ⓘ

Keywords:

Riemann zeta function, approximations, asymptotic

Notes:

See Titchmarsh (1986b, Sections 4.11–4.18).

Permalink:

http://dlmf.nist.gov/25.9

See also:

Annotations for Ch.25

If $x\ge 1$, $y\ge 1$, $2\pi xy=t$, and $0\le \sigma \le 1$, then as $t\to \mathrm{\infty }$ with $\sigma$ fixed,

25.9.1		$$\zeta \left(\sigma +\mathrm{i}t\right)=\sum _{1\le n\le x}\frac{1}{n^s}+\chi (s)\sum _{1\le n\le y}\frac{1}{n^{1-s}}+O\left(x^{-\sigma }\right)+O\left(y^{\sigma -1}{t}^{\frac{1}{2}-\sigma }\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable, $s$: complex variable and $\sigma$: fixed Keywords: asymptotic approximation Source: Titchmarsh (1986b, (4.12.4), p. 79) Referenced by: §25.9 Permalink: http://dlmf.nist.gov/25.9.E1 Encodings: TeX, pMML, png See also: Annotations for §25.9 and Ch.25

where $s=\sigma +\mathrm{i}t$ and

25.9.2		$$\chi (s)\equiv {\pi }^{s-\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}-\frac{1}{2}s\right)/\mathrm{\Gamma }\left(\frac{1}{2}s\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition and $s$: complex variable Keywords: definition Source: Titchmarsh (1986b, p. 78); with (5.5.5), (5.5.3) Referenced by: (25.10.2) Permalink: http://dlmf.nist.gov/25.9.E2 Encodings: TeX, pMML, png See also: Annotations for §25.9 and Ch.25

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

25.9.3		$$\zeta \left(\frac{1}{2}+\mathrm{i}t\right)=\sum _{n=1}^m\frac{1}{n^{\frac{1}{2}+\mathrm{i}t}}+\chi \left(\frac{1}{2}+\mathrm{i}t\right)\sum _{n=1}^m\frac{1}{n^{\frac{1}{2}-\mathrm{i}t}}+O\left(t^{-1/4}\right).$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer and $n$: nonnegative integer Keywords: asymptotic approximation Source: Titchmarsh (1986b, (4.17.1), p. 88) Referenced by: §25.10(ii) Permalink: http://dlmf.nist.gov/25.9.E3 Encodings: TeX, pMML, png See also: Annotations for §25.9 and Ch.25

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

25.8 Sums25.10 Zeros

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