§25.9 Asymptotic Approximations

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

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\mathrm{i}$ : imaginary unit, : nonnegative integer, : real variable, : 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 $s=\sigma+it$ and

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\equiv$ : equals by definition and : 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 $m=\left\lfloor x\right\rfloor$ , then (25.9.1) becomes

25.9.3 $\zeta\left(\tfrac{1}{2}+it\right)=\sum_{n=1}^{m}\frac{1}{n^{\frac{1}{2}+it}}+% \chi\left(\tfrac{1}{2}+it\right)\sum_{n=1}^{m}\frac{1}{n^{\frac{1}{2}-it}}+O% \left(t^{-1/4}\right).$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\mathrm{i}$ : imaginary unit, : nonnegative integer and : 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 Sums 25.10 Zeros

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