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))))))

Fungrim entry: 4c3678

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