Fungrim entry: c31c10

$L\!\left(s, \chi\right) = \frac{1}{{q}^{s}} \sum_{k=1}^{q} \chi(k) \zeta\!\left(s, \frac{k}{q}\right)$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \setminus \left\{1\right\}$
TeX:
L\!\left(s, \chi\right) = \frac{1}{{q}^{s}} \sum_{k=1}^{q} \chi(k) \zeta\!\left(s, \frac{k}{q}\right)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \setminus \left\{1\right\}
Definitions:
Fungrim symbol Notation Short description
DirichletL$L\!\left(s, \chi\right)$ Dirichlet L-function
Pow${a}^{b}$ Power
Sum$\sum_{n} f(n)$ Sum
HurwitzZeta$\zeta\!\left(s, a\right)$ Hurwitz zeta function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
DirichletGroup$G_{q}$ Dirichlet characters with given modulus
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("c31c10"),
Formula(Equal(DirichletL(s, chi), Mul(Div(1, Pow(q, s)), Sum(Mul(chi(k), HurwitzZeta(s, Div(k, q))), For(k, 1, q))))),
Variables(q, chi, s),
Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(s, SetMinus(CC, Set(1))))))

Topics using this entry

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

2020-08-27 09:56:25.682319 UTC