Fungrim entry: 4c3678

$\zeta\!\left(s, \frac{k}{q}\right) = \frac{{q}^{s}}{\varphi(q)} \sum_{\chi \in G_{q}} \overline{\chi(k)} L\!\left(s, \chi\right)$
Assumptions:$q \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; k \in \{1, 2, \ldots, q - 1\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(k, q\right) = 1 \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \setminus \left\{1\right\}$
TeX:
\zeta\!\left(s, \frac{k}{q}\right) = \frac{{q}^{s}}{\varphi(q)} \sum_{\chi \in G_{q}} \overline{\chi(k)} L\!\left(s, \chi\right)

q \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; k \in \{1, 2, \ldots, q - 1\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(k, q\right) = 1 \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \setminus \left\{1\right\}
Definitions:
Fungrim symbol Notation Short description
HurwitzZeta$\zeta\!\left(s, a\right)$ Hurwitz zeta function
Pow${a}^{b}$ Power
Totient$\varphi(n)$ Euler totient function
Sum$\sum_{n} f(n)$ Sum
Conjugate$\overline{z}$ Complex conjugate
DirichletL$L\!\left(s, \chi\right)$ Dirichlet L-function
DirichletGroup$G_{q}$ Dirichlet characters with given modulus
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Range$\{a, a + 1, \ldots, b\}$ Integers between given endpoints
GCD$\gcd\!\left(a, b\right)$ Greatest common divisor
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("4c3678"),
Formula(Equal(HurwitzZeta(s, Div(k, q)), Mul(Div(Pow(q, s), Totient(q)), Sum(Mul(Conjugate(chi(k)), DirichletL(s, chi)), ForElement(chi, DirichletGroup(q)))))),
Variables(q, k, s),
Assumptions(And(Element(q, ZZGreaterEqual(2)), Element(k, Range(1, Sub(q, 1))), Equal(GCD(k, q), 1), 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.

2020-04-08 16:14:44.404316 UTC