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))))))

Fungrim entry: c31c10

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