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Fungrim entry: 69a1a9

ζ ⁣(1s,pq)=2Γ(s)(2πq)sk=1qcos ⁣(πs22πkpq)ζ ⁣(s,kq)\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:sC  and  sZ  and  qZ1  and  p{1,2,,q}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
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
GammaΓ(z)\Gamma(z) Gamma function
Powab{a}^{b} Power
Piπ\pi The constant pi (3.14...)
Sumnf(n)\sum_{n} f(n) Sum
Coscos(z)\cos(z) Cosine
CCC\mathbb{C} Complex numbers
ZZZ\mathbb{Z} Integers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Range{a,a+1,,b}\{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)))))

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2020-08-27 09:56:25.682319 UTC