Fungrim entry: 6c3523

$\zeta\!\left(s, \frac{1}{2} + n\right) = \left({2}^{s} - 1\right) \zeta\!\left(s\right) - {2}^{s} \sum_{k=0}^{n - 1} \frac{1}{{\left(2 k + 1\right)}^{s}}$
Assumptions:$s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}$
TeX:
\zeta\!\left(s, \frac{1}{2} + n\right) = \left({2}^{s} - 1\right) \zeta\!\left(s\right) - {2}^{s} \sum_{k=0}^{n - 1} \frac{1}{{\left(2 k + 1\right)}^{s}}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
HurwitzZeta$\zeta\!\left(s, a\right)$ Hurwitz zeta function
Pow${a}^{b}$ Power
RiemannZeta$\zeta\!\left(s\right)$ Riemann zeta function
Sum$\sum_{n} f(n)$ Sum
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("6c3523"),
Formula(Equal(HurwitzZeta(s, Add(Div(1, 2), n)), Sub(Mul(Sub(Pow(2, s), 1), RiemannZeta(s)), Mul(Pow(2, s), Sum(Div(1, Pow(Add(Mul(2, k), 1), s)), For(k, 0, Sub(n, 1))))))),
Variables(s, n),
Assumptions(And(Element(s, CC), Element(n, ZZGreaterEqual(0)))))

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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