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Fungrim entry: 7d9feb

ζ ⁣(s,a)=1Nsk=0N1ζ ⁣(s,a+kN)\zeta\!\left(s, a\right) = \frac{1}{{N}^{s}} \sum_{k=0}^{N - 1} \zeta\!\left(s, \frac{a + k}{N}\right)
Assumptions:sC  and  aC  and  s1  and  Re(a)>0  and  NZ1s \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, a\right) = \frac{1}{{N}^{s}} \sum_{k=0}^{N - 1} \zeta\!\left(s, \frac{a + 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
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
Powab{a}^{b} Power
Sumnf(n)\sum_{n} f(n) Sum
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("7d9feb"),
    Formula(Equal(HurwitzZeta(s, a), Mul(Div(1, Pow(N, s)), Sum(HurwitzZeta(s, Div(Add(a, 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)))))

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2020-08-27 09:56:25.682319 UTC