# Fungrim entry: 692e42

$\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} \zeta\!\left(s\right) = \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\} \cup \left\{ \rho_{n} : n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0 \right\}$
TeX:
\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\}
Definitions:
Fungrim symbol Notation Short description
Zeros$\mathop{\operatorname{zeros}\,}\limits_{P\left(x\right)} f\!\left(x\right)$ Zeros (roots) of function
RiemannZeta$\zeta\!\left(s\right)$ Riemann zeta function
CC$\mathbb{C}$ Complex numbers
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
RiemannZetaZero$\rho_{n}$ Nontrivial zero of the Riemann zeta function
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("692e42"),
Formula(Equal(Zeros(RiemannZeta(s), s, Element(s, CC)), Union(SetBuilder(Neg(Mul(2, n)), n, Element(n, ZZGreaterEqual(1))), SetBuilder(RiemannZetaZero(n), n, And(Element(n, ZZ), Unequal(n, 0)))))))

## Topics using this entry

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

2019-08-17 11:32:46.829430 UTC