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
`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)))))))

Fungrim entry: 692e42

Topics using this entry