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Fungrim entry: c31c10

L ⁣(s,χ)=1qsk=1qχ(k)ζ ⁣(s,kq)L\!\left(s, \chi\right) = \frac{1}{{q}^{s}} \sum_{k=1}^{q} \chi(k) \zeta\!\left(s, \frac{k}{q}\right)
Assumptions:qZ1  and  χGq  and  sC{1}q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \setminus \left\{1\right\}
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\}
Fungrim symbol Notation Short description
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
Powab{a}^{b} Power
Sumnf(n)\sum_{n} f(n) Sum
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
CCC\mathbb{C} Complex numbers
Source code for this entry:
    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))))))

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

2021-03-15 19:12:00.328586 UTC