Fungrim home page

Fungrim entry: 4c3678

ζ ⁣(s,kq)=qsφ(q)χGqχ(k)L ⁣(s,χ)\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:qZ2andk{1,2,,q1}andgcd ⁣(k,q)=1andsC{1}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
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
Powab{a}^{b} Power
Totientφ(n)\varphi(n) Euler totient function
Sumnf(n)\sum_{n} f(n) Sum
Conjugatez\overline{z} Complex conjugate
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
DirichletGroupGqG_{q} Dirichlet characters with given modulus
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
CCC\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.

2019-10-05 13:11:19.856591 UTC