Fungrim entry: 982e3b

0 < \operatorname{Re}\!\left(\rho_{n,\chi}\right) < 1

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G^{\text{Primitive}}_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 0

TeX:

0 < \operatorname{Re}\!\left(\rho_{n,\chi}\right) < 1

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G^{\text{Primitive}}_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Re`	$\operatorname{Re}(z)$	Real part
`DirichletLZero`	$\rho_{n,\chi}$	Nontrivial zero of Dirichlet L-function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("982e3b"),
    Formula(Less(0, Re(DirichletLZero(n, chi)), 1)),
    Variables(q, n, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, PrimitiveDirichletCharacters(q)), Element(n, ZZ), NotEqual(n, 0))))

Topics using this entry

Dirichlet L-functions