# Riemann zeta function

## Definitions

Symbol: RiemannZeta $\zeta\!\left(s\right)$ Riemann zeta function

## Illustrations

Image: X-ray of $\zeta\!\left(s\right)$ on $s \in \left[-22, 22\right] + \left[-27, 27\right] i$ with the critical strip highlighted

## Dirichlet series

$\zeta\!\left(s\right) = \sum_{k=1}^{\infty} \frac{1}{{k}^{s}}$
$\frac{1}{\zeta\!\left(s\right)} = \sum_{k=1}^{\infty} \frac{\mu(k)}{{k}^{s}}$

## Euler product

$\zeta\!\left(s\right) = \prod_{p} \frac{1}{1 - \frac{1}{{p}^{s}}}$

## Laurent series

Related topic: Stieltjes constants
$\zeta\!\left(s\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n} {\left(s - 1\right)}^{n}$

## Special values

$\zeta\!\left(2\right) = \frac{{\pi}^{2}}{6}$
$\zeta\!\left(3\right) \notin \mathbb{Q}$
$\zeta\!\left(2 n\right) = \frac{{\left(-1\right)}^{n + 1} B_{2 n} {\left(2 \pi\right)}^{2 n}}{2 \left(2 n\right)!}$
$\zeta\!\left(-n\right) = \frac{{\left(-1\right)}^{n} B_{n + 1}}{n + 1}$
Table of $\zeta\!\left(2 n\right)$ for $1 \le n \le 20$
Table of $\zeta\!\left(-n\right)$ for $0 \le n \le 30$
Table of $\zeta\!\left(n\right)$ to 50 digits for $2 \le n \le 50$

## Analytic properties

$\zeta\!\left(s\right) \text{ is holomorphic on } s \in \mathbb{C} \setminus \left\{1\right\}$
$\mathop{\operatorname{poles}\,}\limits_{s \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} \zeta\!\left(s\right) = \left\{1\right\}$
$\operatorname{EssentialSingularities}\!\left(\zeta\!\left(s\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}$
$\operatorname{BranchPoints}\!\left(\zeta\!\left(s\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{BranchCuts}\!\left(\zeta\!\left(s\right), s, \mathbb{C}\right) = \left\{\right\}$

## Zeros

Related topics: Zeros of the Riemann zeta function

Symbol: RiemannZetaZero $\rho_{n}$ Nontrivial zero of the Riemann zeta function
Symbol: RiemannHypothesis $\operatorname{RH}$ Riemann hypothesis
$\left(\operatorname{RH}\right) \iff \left(\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\right)$
$\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{R}} \zeta\!\left(s\right) = \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\}$
$\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,0 \le \operatorname{Re}(s) \le 1} \zeta\!\left(s\right) = \left\{ \rho_{n} : n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 0 \right\}$
$\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} \zeta\!\left(s\right) = \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\} \cup \left\{ \rho_{n} : n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 0 \right\}$
$0 < \operatorname{Re}\!\left(\rho_{n}\right) < 1$
$\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2}$
$\rho_{-n} = \overline{\rho_{n}}$
Table of $\operatorname{Im}\!\left(\rho_{n}\right)$ to 50 digits for $1 \le n \le 50$

## Complex parts

$\zeta\!\left(\overline{s}\right) = \overline{\zeta\!\left(s\right)}$

## Functional equation

$\zeta\!\left(s\right) = 2 {\left(2 \pi\right)}^{s - 1} \sin\!\left(\frac{\pi s}{2}\right) \Gamma\!\left(1 - s\right) \zeta\!\left(1 - s\right)$
$\zeta\!\left(1 - s\right) = \frac{2 \cos\!\left(\frac{1}{2} \pi s\right)}{{\left(2 \pi\right)}^{s}} \Gamma(s) \zeta\!\left(s\right)$

## Bounds and inequalities

$\left|\zeta\!\left(s\right)\right| \le \zeta\!\left(\operatorname{Re}(s)\right)$
$\left|\zeta\!\left(s\right)\right| < 3 \left|\frac{1 + s}{1 - s}\right| {\left|\frac{1 + s}{2 \pi}\right|}^{\left( 1 + \eta - \operatorname{Re}(s) \right) / 2} \zeta\!\left(1 + \eta\right)$

## Euler-Maclaurin formula

$\zeta\!\left(s\right) = \sum_{k=1}^{N - 1} \frac{1}{{k}^{s}} + \frac{{N}^{1 - s}}{s - 1} + \frac{1}{{N}^{s}} \left(\frac{1}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \frac{\left(s\right)_{2 k - 1}}{{N}^{2 k - 1}}\right) - \int_{N}^{\infty} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} \frac{\left(s\right)_{2 M}}{{t}^{s + 2 M}} \, dt$

## Approximations

$\left|\zeta\!\left(s\right) - \left(\sum_{k=1}^{N - 1} \frac{1}{{k}^{s}} + \frac{{N}^{1 - s}}{s - 1} + \frac{1}{{N}^{s}} \left(\frac{1}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \frac{\left(s\right)_{2 k - 1}}{{N}^{2 k - 1}}\right)\right)\right| \le \frac{4 \left|\left(s\right)_{2 M}\right|}{{\left(2 \pi\right)}^{2 M}} \frac{{N}^{-\left(\operatorname{Re}(s) + 2 M - 1\right)}}{\operatorname{Re}(s) + 2 M - 1}$
$\left|\left(1 - {2}^{1 - s}\right) \zeta\!\left(s\right) - \frac{1}{d(n)} \sum_{k=0}^{n - 1} \frac{{\left(-1\right)}^{k} \left(d(n) - d(k)\right)}{{\left(k + 1\right)}^{s}}\right| \le \frac{3 \left(1 + 2 \left|\operatorname{Im}(s)\right|\right)}{{\left(3 + \sqrt{8}\right)}^{n}} {e}^{\left|\operatorname{Im}(s)\right| \pi / 2}\; \text{ where } d(k) = n \sum_{i=0}^{k} \frac{\left(n + i - 1\right)! {4}^{i}}{\left(n - i\right)! \left(2 i\right)!}$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC