Fungrim entry: ba7f85

\zeta\!\left(s, N a\right) = \frac{1}{{N}^{s}} \sum_{k=0}^{N - 1} \zeta\!\left(s, a + \frac{k}{N}\right)

Assumptions:

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}

TeX:

\zeta\!\left(s, N a\right) = \frac{1}{{N}^{s}} \sum_{k=0}^{N - 1} \zeta\!\left(s, a + \frac{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
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("ba7f85"),
    Formula(Equal(HurwitzZeta(s, Mul(N, a)), Mul(Div(1, Pow(N, s)), Sum(HurwitzZeta(s, Add(a, Div(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)))))

Topics using this entry

Hurwitz zeta function