# Fungrim entry: ba7f85

$\zeta\!\left(s, N a\right) = \frac{1}{{N}^{s}} \sum_{k=0}^{N - 1} \zeta\!\left(s, a + \frac{k}{N}\right)$
Assumptions:$s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}$
TeX:
\zeta\!\left(s, N a\right) = \frac{1}{{N}^{s}} \sum_{k=0}^{N - 1} \zeta\!\left(s, a + \frac{k}{N}\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
HurwitzZeta$\zeta\!\left(s, a\right)$ Hurwitz zeta function
Pow${a}^{b}$ Power
Sum$\sum_{n} f(n)$ Sum
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("ba7f85"),
Formula(Equal(HurwitzZeta(s, Mul(N, a)), Mul(Div(1, Pow(N, s)), Sum(HurwitzZeta(s, Add(a, Div(k, N))), For(k, 0, Sub(N, 1)))))),
Variables(s, a, N),
Assumptions(And(Element(s, CC), Element(a, CC), NotEqual(s, 1), Greater(Re(a), 0), Element(N, ZZGreaterEqual(1)))))

## Topics using this entry

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