# Fungrim entry: bed7ee

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

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{or}}\; s = 0\right) \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
HurwitzZeta$\zeta\!\left(s, a\right)$ Hurwitz zeta function
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
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("bed7ee"),
Formula(Equal(HurwitzZeta(s, Add(a, N)), Sub(HurwitzZeta(s, a), Sum(Div(1, Pow(Add(n, a), s)), For(n, 0, Sub(N, 1)))))),
Variables(s, a, N),
Assumptions(And(Element(s, CC), Element(a, CC), NotEqual(s, 1), Or(NotElement(a, ZZLessEqual(0)), Less(Re(s), 0), Equal(s, 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.

2020-08-27 09:56:25.682319 UTC