# Fungrim entry: 69a1a9

$\zeta\!\left(1 - s, \frac{p}{q}\right) = \frac{2 \Gamma(s)}{{\left(2 \pi q\right)}^{s}} \sum_{k=1}^{q} \cos\!\left(\frac{\pi s}{2} - \frac{2 \pi k p}{q}\right) \zeta\!\left(s, \frac{k}{q}\right)$
Assumptions:$s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \mathbb{Z} \;\mathbin{\operatorname{and}}\; q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; p \in \{1, 2, \ldots, q\}$
TeX:
\zeta\!\left(1 - s, \frac{p}{q}\right) = \frac{2 \Gamma(s)}{{\left(2 \pi q\right)}^{s}} \sum_{k=1}^{q} \cos\!\left(\frac{\pi s}{2} - \frac{2 \pi k p}{q}\right) \zeta\!\left(s, \frac{k}{q}\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \mathbb{Z} \;\mathbin{\operatorname{and}}\; q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; p \in \{1, 2, \ldots, q\}
Definitions:
Fungrim symbol Notation Short description
HurwitzZeta$\zeta\!\left(s, a\right)$ Hurwitz zeta function
Gamma$\Gamma(z)$ Gamma function
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
Sum$\sum_{n} f(n)$ Sum
Cos$\cos(z)$ Cosine
CC$\mathbb{C}$ Complex numbers
ZZ$\mathbb{Z}$ Integers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Range$\{a, a + 1, \ldots, b\}$ Integers between given endpoints
Source code for this entry:
Entry(ID("69a1a9"),
Formula(Equal(HurwitzZeta(Sub(1, s), Div(p, q)), Mul(Div(Mul(2, Gamma(s)), Pow(Mul(Mul(2, Pi), q), s)), Sum(Mul(Cos(Sub(Div(Mul(Pi, s), 2), Div(Mul(Mul(Mul(2, Pi), k), p), q))), HurwitzZeta(s, Div(k, q))), For(k, 1, q))))),
Variables(s, p, q),
Assumptions(And(Element(s, CC), NotElement(s, ZZ), Element(q, ZZGreaterEqual(1)), Element(p, Range(1, q)))))

## 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