Fungrim entry: 69a1a9

\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:

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
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Gamma`	$\Gamma(z)$	Gamma function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Sum`	$\sum_{n} f(n)$	Sum
`Cos`	$\cos(z)$	Cosine
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	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)))))

Topics using this entry

Hurwitz zeta function