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)))))

Topics using this entry

Hurwitz zeta function