Hurwitz zeta function

Symbol: HurwitzZeta —

\zeta\!\left(s, a\right)

— Hurwitz zeta function

►HurwitzZeta(s, a) —

\zeta\!\left(s, a\right)

— represents the Hurwitz zeta function of argument

s

and parameter

a

.

►HurwitzZeta(s, a, 1) —

\zeta'\!\left(s, a\right)

— represents the Hurwitz zeta function of argument

s

and parameter

a

, differentiated once with respect to

s

.

►HurwitzZeta(s, a, r) —

\zeta^{(r)}\!\left(s, a\right)

— represents the Hurwitz zeta function of argument

s

and parameter

a

, differentiated to order

r

with respect to

s

.

References:

https://dlmf.nist.gov/25.11
http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function

Source code for this entry:

Entry(ID("04217b"),
    SymbolDefinition(HurwitzZeta, HurwitzZeta(s, a), "Hurwitz zeta function"),
    CodeExample(HurwitzZeta(s, a), "represents the Hurwitz zeta function of argument", s, "and parameter", a, "."),
    CodeExample(HurwitzZeta(s, a, 1), "represents the Hurwitz zeta function of argument", s, "and parameter", a, ", differentiated once with respect to", s, "."),
    CodeExample(HurwitzZeta(s, a, r), "represents the Hurwitz zeta function of argument", s, "and parameter", a, ", differentiated to order", r, "with respect to", s, "."),
    References("https://dlmf.nist.gov/25.11", "http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/"))

855201

Image: Plot of

\zeta\!\left(s, a\right)

on

s \in \left[-25, 11\right]

for

a \in \left\{0.6, 0.8, 1.4\right\}

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`ClosedInterval`	$\left[a, b\right]$	Closed interval

Source code for this entry:

Entry(ID("855201"),
    Image(Description("Plot of", HurwitzZeta(s, a), "on", Element(s, ClosedInterval(-25, 11)), "for", Element(a, Set(Decimal("0.6"), Decimal("0.8"), Decimal("1.4")))), ImageSource("plot_hurwitz_zeta")))

583bf9

Image: X-ray of

\zeta\!\left(s, 1 + \frac{i}{2}\right)

on

s \in \left[-20, 20\right] + \left[-20, 20\right] i

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f(z)

. Thick black curves show where

\operatorname{Im}\!\left(f(z)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f(z)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f(z)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`ConstI`	$i$	Imaginary unit
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("583bf9"),
    Image(Description("X-ray of", HurwitzZeta(s, Add(1, Div(ConstI, 2))), "on", Element(s, Add(ClosedInterval(-20, 20), Mul(ClosedInterval(-20, 20), ConstI)))), ImageSource("xray_hurwitz_zeta")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

0e2bcb

Image: X-ray of

\zeta\!\left(2 + 3 i, a\right)

on

a \in \left[-5, 5\right] + \left[-5, 5\right] i

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f(z)

. Thick black curves show where

\operatorname{Im}\!\left(f(z)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f(z)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f(z)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`ConstI`	$i$	Imaginary unit
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("0e2bcb"),
    Image(Description("X-ray of", HurwitzZeta(Add(2, Mul(3, ConstI)), a), "on", Element(a, Add(ClosedInterval(-5, 5), Mul(ClosedInterval(-5, 5), ConstI)))), ImageSource("xray_hurwitz_zeta_param")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

56dcbd

\left(s \in \mathbb{C} \setminus \left\{1\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{C}

TeX:

\left(s \in \mathbb{C} \setminus \left\{1\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function

Source code for this entry:

Entry(ID("56dcbd"),
    Formula(Implies(And(Element(s, SetMinus(CC, Set(1))), Element(a, SetMinus(CC, ZZLessEqual(0)))), Element(HurwitzZeta(s, a), CC))),
    Variables(s, a))

ad269f

\left(s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{C}

TeX:

\left(s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function

Source code for this entry:

Entry(ID("ad269f"),
    Formula(Implies(And(Element(s, CC), Less(Re(s), 0), Element(a, CC)), Element(HurwitzZeta(s, a), CC))),
    Variables(s, a))

e7224b

\left(s \in \mathbb{R} \setminus \left\{1\right\} \;\mathbin{\operatorname{and}}\; a \in \left(0, \infty\right)\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{R}

TeX:

\left(s \in \mathbb{R} \setminus \left\{1\right\} \;\mathbin{\operatorname{and}}\; a \in \left(0, \infty\right)\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`RR`	$\mathbb{R}$	Real numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function

Source code for this entry:

Entry(ID("e7224b"),
    Formula(Implies(And(Element(s, SetMinus(RR, Set(1))), Element(a, OpenInterval(0, Infinity))), Element(HurwitzZeta(s, a), RR))),
    Variables(s, a))

b0f500

\left(s \in \mathbb{Z} \setminus \left\{1\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{R} \setminus \{0, -1, \ldots\}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{R}

TeX:

\left(s \in \mathbb{Z} \setminus \left\{1\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{R} \setminus \{0, -1, \ldots\}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`ZZ`	$\mathbb{Z}$	Integers
`RR`	$\mathbb{R}$	Real numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function

Source code for this entry:

Entry(ID("b0f500"),
    Formula(Implies(And(Element(s, SetMinus(ZZ, Set(1))), Element(a, SetMinus(RR, ZZLessEqual(0)))), Element(HurwitzZeta(s, a), RR))),
    Variables(s, a))

cc523f

\left(s \in \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{Q}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{Q}

TeX:

\left(s \in \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{Q}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{Q}

Definitions:

Fungrim symbol	Notation	Short description
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`QQ`	$\mathbb{Q}$	Rational numbers
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function

Source code for this entry:

Entry(ID("cc523f"),
    Formula(Implies(And(Element(s, ZZLessEqual(0)), Element(a, QQ)), Element(HurwitzZeta(s, a), QQ))),
    Variables(s, a))

ec2dd5

\left(s \in \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{R}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{R}

TeX:

\left(s \in \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{R}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`RR`	$\mathbb{R}$	Real numbers
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function

Source code for this entry:

Entry(ID("ec2dd5"),
    Formula(Implies(And(Element(s, ZZLessEqual(0)), Element(a, RR)), Element(HurwitzZeta(s, a), RR))),
    Variables(s, a))

a5980a

\left(s \in \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{C}

TeX:

\left(s \in \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function

Source code for this entry:

Entry(ID("a5980a"),
    Formula(Implies(And(Element(s, ZZLessEqual(0)), Element(a, CC)), Element(HurwitzZeta(s, a), CC))),
    Variables(s, a))

d0b234

\left(s \in \left\{1\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right) \;\implies\; \zeta\!\left(s, a\right) \in \left\{{\tilde \infty}\right\}

TeX:

\left(s \in \left\{1\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right) \;\implies\; \zeta\!\left(s, a\right) \in \left\{{\tilde \infty}\right\}

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("d0b234"),
    Formula(Implies(And(Element(s, Set(1)), Element(a, SetMinus(CC, ZZLessEqual(0)))), Element(HurwitzZeta(s, a), Set(UnsignedInfinity)))),
    Variables(s, a))

c5d844

\left(s \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; a \in \{0, -1, \ldots\}\right) \;\implies\; \zeta\!\left(s, a\right) \in \left\{{\tilde \infty}\right\}

TeX:

\left(s \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; a \in \{0, -1, \ldots\}\right) \;\implies\; \zeta\!\left(s, a\right) \in \left\{{\tilde \infty}\right\}

Definitions:

Fungrim symbol	Notation	Short description
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("c5d844"),
    Formula(Implies(And(Element(s, ZZGreaterEqual(2)), Element(a, ZZLessEqual(0))), Element(HurwitzZeta(s, a), Set(UnsignedInfinity)))),
    Variables(s, a))

4bf3da

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } s \in \mathbb{C} \setminus \left\{1\right\}\right)

TeX:

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } s \in \mathbb{C} \setminus \left\{1\right\}\right)

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function

Source code for this entry:

Entry(ID("4bf3da"),
    Formula(Implies(Element(a, SetMinus(CC, ZZLessEqual(0))), IsHolomorphic(HurwitzZeta(s, a), ForElement(s, SetMinus(CC, Set(1)))))),
    Variables(a))

ea271f

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is meromorphic on } s \in \mathbb{C}\right)

TeX:

a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is meromorphic on } s \in \mathbb{C}\right)

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`IsMeromorphic`	$f(z) \text{ is meromorphic at } z = c$	Meromorphic predicate
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function

Source code for this entry:

Entry(ID("ea271f"),
    Formula(Implies(Element(a, SetMinus(CC, ZZLessEqual(0))), IsMeromorphic(HurwitzZeta(s, a), ForElement(s, CC)))),
    Variables(a))

26418b

a \in \mathbb{C} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } s \in \left\{ t : t \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(t) < 0 \right\}\right)

TeX:

a \in \mathbb{C} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } s \in \left\{ t : t \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(t) < 0 \right\}\right)

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("26418b"),
    Formula(Implies(Element(a, CC), IsHolomorphic(HurwitzZeta(s, a), ForElement(s, Set(t, ForElement(t, CC), Less(Re(t), 0)))))),
    Variables(a))

93e149

s \in \mathbb{C} \setminus \left\{1\right\} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C} \setminus \left(-\infty, 0\right]\right)

TeX:

s \in \mathbb{C} \setminus \left\{1\right\} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C} \setminus \left(-\infty, 0\right]\right)

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("93e149"),
    Formula(Implies(Element(s, SetMinus(CC, Set(1))), IsHolomorphic(HurwitzZeta(s, a), ForElement(a, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))))),
    Variables(s))

8c7cdb

s \in \mathbb{Z}_{\ge 2} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right)

TeX:

s \in \mathbb{Z}_{\ge 2} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right)

Definitions:

Fungrim symbol	Notation	Short description
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("8c7cdb"),
    Formula(Implies(Element(s, ZZGreaterEqual(2)), IsHolomorphic(HurwitzZeta(s, a), ForElement(a, SetMinus(CC, ZZLessEqual(0)))))),
    Variables(s))

05c2dd

s \in \mathbb{Z}_{\ge 2} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is meromorphic on } a \in \mathbb{C}\right)

TeX:

s \in \mathbb{Z}_{\ge 2} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is meromorphic on } a \in \mathbb{C}\right)

Definitions:

Fungrim symbol	Notation	Short description
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`IsMeromorphic`	$f(z) \text{ is meromorphic at } z = c$	Meromorphic predicate
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("05c2dd"),
    Formula(Implies(Element(s, ZZGreaterEqual(2)), IsMeromorphic(HurwitzZeta(s, a), ForElement(a, CC)))),
    Variables(s))

f045b3

s \in \{0, -1, \ldots\} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C}\right)

TeX:

s \in \{0, -1, \ldots\} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C}\right)

Definitions:

Fungrim symbol	Notation	Short description
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f045b3"),
    Formula(Implies(Element(s, ZZLessEqual(0)), IsHolomorphic(HurwitzZeta(s, a), ForElement(a, CC)))),
    Variables(s))

af23f7

\zeta\!\left(s, 1\right) = \zeta\!\left(s\right)

Assumptions:

s \in \mathbb{C}

TeX:

\zeta\!\left(s, 1\right) = \zeta\!\left(s\right)

s \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("af23f7"),
    Formula(Equal(HurwitzZeta(s, 1), RiemannZeta(s))),
    Variables(s),
    Assumptions(Element(s, CC)))

b721b4

\zeta\!\left(s, 2\right) = \zeta\!\left(s\right) - 1

Assumptions:

s \in \mathbb{C}

TeX:

\zeta\!\left(s, 2\right) = \zeta\!\left(s\right) - 1

s \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b721b4"),
    Formula(Equal(HurwitzZeta(s, 2), Sub(RiemannZeta(s), 1))),
    Variables(s),
    Assumptions(Element(s, CC)))

fc6fe0

\zeta\!\left(s, 3\right) = \zeta\!\left(s\right) - 1 - \frac{1}{{2}^{s}}

Assumptions:

s \in \mathbb{C}

TeX:

\zeta\!\left(s, 3\right) = \zeta\!\left(s\right) - 1 - \frac{1}{{2}^{s}}

s \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("fc6fe0"),
    Formula(Equal(HurwitzZeta(s, 3), Sub(Sub(RiemannZeta(s), 1), Div(1, Pow(2, s))))),
    Variables(s),
    Assumptions(Element(s, CC)))

6e69fc

\zeta\!\left(s, n\right) = \zeta\!\left(s\right) - \sum_{k=1}^{n - 1} \frac{1}{{k}^{s}}

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

TeX:

\zeta\!\left(s, n\right) = \zeta\!\left(s\right) - \sum_{k=1}^{n - 1} \frac{1}{{k}^{s}}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("6e69fc"),
    Formula(Equal(HurwitzZeta(s, n), Sub(RiemannZeta(s), Sum(Div(1, Pow(k, s)), For(k, 1, Sub(n, 1)))))),
    Variables(s, n),
    Assumptions(And(Element(s, CC), Element(n, ZZGreaterEqual(1)))))

af7d3d

\zeta\!\left(s, \frac{1}{2}\right) = \left({2}^{s} - 1\right) \zeta\!\left(s\right)

Assumptions:

s \in \mathbb{C}

TeX:

\zeta\!\left(s, \frac{1}{2}\right) = \left({2}^{s} - 1\right) \zeta\!\left(s\right)

s \in \mathbb{C}

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
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("af7d3d"),
    Formula(Equal(HurwitzZeta(s, Div(1, 2)), Mul(Sub(Pow(2, s), 1), RiemannZeta(s)))),
    Variables(s),
    Assumptions(Element(s, CC)))

c6d6e2

\zeta\!\left(s, \frac{3}{2}\right) = \left({2}^{s} - 1\right) \zeta\!\left(s\right) - {2}^{s}

Assumptions:

s \in \mathbb{C}

TeX:

\zeta\!\left(s, \frac{3}{2}\right) = \left({2}^{s} - 1\right) \zeta\!\left(s\right) - {2}^{s}

s \in \mathbb{C}

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
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("c6d6e2"),
    Formula(Equal(HurwitzZeta(s, Div(3, 2)), Sub(Mul(Sub(Pow(2, s), 1), RiemannZeta(s)), Pow(2, s)))),
    Variables(s),
    Assumptions(Element(s, CC)))

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)))))

8bbb6f

\zeta\!\left(s, \frac{1}{4}\right) + \zeta\!\left(s, \frac{3}{4}\right) = {2}^{s} \left({2}^{s} - 1\right) \zeta\!\left(s\right)

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1

TeX:

\zeta\!\left(s, \frac{1}{4}\right) + \zeta\!\left(s, \frac{3}{4}\right) = {2}^{s} \left({2}^{s} - 1\right) \zeta\!\left(s\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1

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
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("8bbb6f"),
    Formula(Equal(Add(HurwitzZeta(s, Div(1, 4)), HurwitzZeta(s, Div(3, 4))), Mul(Mul(Pow(2, s), Sub(Pow(2, s), 1)), RiemannZeta(s)))),
    Variables(s),
    Assumptions(And(Element(s, CC), NotEqual(s, 1))))

4d1f6b

\zeta\!\left(s, \frac{1}{6}\right) + \zeta\!\left(s, \frac{5}{6}\right) = \left({2}^{s} - 1\right) \left({3}^{s} - 1\right) \zeta\!\left(s\right)

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1

TeX:

\zeta\!\left(s, \frac{1}{6}\right) + \zeta\!\left(s, \frac{5}{6}\right) = \left({2}^{s} - 1\right) \left({3}^{s} - 1\right) \zeta\!\left(s\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1

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
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4d1f6b"),
    Formula(Equal(Add(HurwitzZeta(s, Div(1, 6)), HurwitzZeta(s, Div(5, 6))), Mul(Mul(Sub(Pow(2, s), 1), Sub(Pow(3, s), 1)), RiemannZeta(s)))),
    Variables(s),
    Assumptions(And(Element(s, CC), NotEqual(s, 1))))

575b8f

\zeta\!\left(2, 1\right) = \frac{{\pi}^{2}}{6}

TeX:

\zeta\!\left(2, 1\right) = \frac{{\pi}^{2}}{6}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("575b8f"),
    Formula(Equal(HurwitzZeta(2, 1), Div(Pow(Pi, 2), 6))))

ac8d3c

\zeta\!\left(2, 2\right) = \frac{{\pi}^{2}}{6} - 1

TeX:

\zeta\!\left(2, 2\right) = \frac{{\pi}^{2}}{6} - 1

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("ac8d3c"),
    Formula(Equal(HurwitzZeta(2, 2), Sub(Div(Pow(Pi, 2), 6), 1))))

b4ed44

\zeta\!\left(3, 1\right) = \zeta\!\left(3\right)

TeX:

\zeta\!\left(3, 1\right) = \zeta\!\left(3\right)

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function

Source code for this entry:

Entry(ID("b4ed44"),
    Formula(Equal(HurwitzZeta(3, 1), RiemannZeta(3))))

4dd87c

\zeta\!\left(3, 2\right) = \zeta\!\left(3\right) - 1

TeX:

\zeta\!\left(3, 2\right) = \zeta\!\left(3\right) - 1

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function

Source code for this entry:

Entry(ID("4dd87c"),
    Formula(Equal(HurwitzZeta(3, 2), Sub(RiemannZeta(3), 1))))

2d4828

\zeta\!\left(4, 1\right) = \frac{{\pi}^{4}}{90}

TeX:

\zeta\!\left(4, 1\right) = \frac{{\pi}^{4}}{90}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("2d4828"),
    Formula(Equal(HurwitzZeta(4, 1), Div(Pow(Pi, 4), 90))))

33690e

\zeta\!\left(4, 2\right) = \frac{{\pi}^{4}}{90} - 1

TeX:

\zeta\!\left(4, 2\right) = \frac{{\pi}^{4}}{90} - 1

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("33690e"),
    Formula(Equal(HurwitzZeta(4, 2), Sub(Div(Pow(Pi, 4), 90), 1))))

868061

\zeta\!\left(2, \frac{1}{2}\right) = \frac{{\pi}^{2}}{2}

TeX:

\zeta\!\left(2, \frac{1}{2}\right) = \frac{{\pi}^{2}}{2}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("868061"),
    Formula(Equal(HurwitzZeta(2, Div(1, 2)), Div(Pow(Pi, 2), 2))))

9417f4

\zeta\!\left(3, \frac{1}{2}\right) = 7 \zeta\!\left(3\right)

TeX:

\zeta\!\left(3, \frac{1}{2}\right) = 7 \zeta\!\left(3\right)

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function

Source code for this entry:

Entry(ID("9417f4"),
    Formula(Equal(HurwitzZeta(3, Div(1, 2)), Mul(7, RiemannZeta(3)))))

4064f5

\zeta\!\left(4, \frac{1}{2}\right) = \frac{{\pi}^{4}}{6}

TeX:

\zeta\!\left(4, \frac{1}{2}\right) = \frac{{\pi}^{4}}{6}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("4064f5"),
    Formula(Equal(HurwitzZeta(4, Div(1, 2)), Div(Pow(Pi, 4), 6))))

3e82c3

\zeta\!\left(2, \frac{1}{4}\right) = {\pi}^{2} + 8 G

TeX:

\zeta\!\left(2, \frac{1}{4}\right) = {\pi}^{2} + 8 G

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ConstCatalan`	$G$	Catalan's constant

Source code for this entry:

Entry(ID("3e82c3"),
    Formula(Equal(HurwitzZeta(2, Div(1, 4)), Add(Pow(Pi, 2), Mul(8, ConstCatalan)))))

951f86

\zeta\!\left(2, \frac{3}{4}\right) = {\pi}^{2} - 8 G

TeX:

\zeta\!\left(2, \frac{3}{4}\right) = {\pi}^{2} - 8 G

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ConstCatalan`	$G$	Catalan's constant

Source code for this entry:

Entry(ID("951f86"),
    Formula(Equal(HurwitzZeta(2, Div(3, 4)), Sub(Pow(Pi, 2), Mul(8, ConstCatalan)))))

eda0f3

\zeta\!\left(3, \frac{1}{4}\right) = 28 \zeta\!\left(3\right) + {\pi}^{3}

TeX:

\zeta\!\left(3, \frac{1}{4}\right) = 28 \zeta\!\left(3\right) + {\pi}^{3}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("eda0f3"),
    Formula(Equal(HurwitzZeta(3, Div(1, 4)), Add(Mul(28, RiemannZeta(3)), Pow(Pi, 3)))))

b347d3

\zeta\!\left(3, \frac{3}{4}\right) = 28 \zeta\!\left(3\right) - {\pi}^{3}

TeX:

\zeta\!\left(3, \frac{3}{4}\right) = 28 \zeta\!\left(3\right) - {\pi}^{3}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("b347d3"),
    Formula(Equal(HurwitzZeta(3, Div(3, 4)), Sub(Mul(28, RiemannZeta(3)), Pow(Pi, 3)))))

2fabeb

\zeta\!\left(3, \frac{1}{6}\right) = 91 \zeta\!\left(3\right) + 2 \sqrt{3} {\pi}^{3}

TeX:

\zeta\!\left(3, \frac{1}{6}\right) = 91 \zeta\!\left(3\right) + 2 \sqrt{3} {\pi}^{3}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("2fabeb"),
    Formula(Equal(HurwitzZeta(3, Div(1, 6)), Add(Mul(91, RiemannZeta(3)), Mul(Mul(2, Sqrt(3)), Pow(Pi, 3))))))

edad97

\zeta\!\left(3, \frac{5}{6}\right) = 91 \zeta\!\left(3\right) - 2 \sqrt{3} {\pi}^{3}

TeX:

\zeta\!\left(3, \frac{5}{6}\right) = 91 \zeta\!\left(3\right) - 2 \sqrt{3} {\pi}^{3}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("edad97"),
    Formula(Equal(HurwitzZeta(3, Div(5, 6)), Sub(Mul(91, RiemannZeta(3)), Mul(Mul(2, Sqrt(3)), Pow(Pi, 3))))))

84196a

\zeta\!\left(n, a\right) = \frac{{\left(-1\right)}^{n}}{\left(n - 1\right)!} \psi^{(n - 1)}\!\left(a\right)

Assumptions:

n \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C}

TeX:

\zeta\!\left(n, a\right) = \frac{{\left(-1\right)}^{n}}{\left(n - 1\right)!} \psi^{(n - 1)}\!\left(a\right)

n \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("84196a"),
    Formula(Equal(HurwitzZeta(n, a), Mul(Div(Pow(-1, n), Factorial(Sub(n, 1))), DigammaFunction(a, Sub(n, 1))))),
    Variables(n, a),
    Assumptions(And(Element(n, ZZGreaterEqual(2)), Element(a, CC))))

532f31

\zeta\!\left(1, a\right) = {\tilde \infty}

Assumptions:

a \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\zeta\!\left(1, a\right) = {\tilde \infty}

a \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("532f31"),
    Formula(Equal(HurwitzZeta(1, a), UnsignedInfinity)),
    Variables(a),
    Assumptions(Element(a, SetMinus(CC, ZZLessEqual(0)))))

5bdba2

\zeta\!\left(-n, a\right) = -\frac{B_{n + 1}\!\left(a\right)}{n + 1}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C}

TeX:

\zeta\!\left(-n, a\right) = -\frac{B_{n + 1}\!\left(a\right)}{n + 1}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("5bdba2"),
    Formula(Equal(HurwitzZeta(Neg(n), a), Neg(Div(BernoulliPolynomial(Add(n, 1), a), Add(n, 1))))),
    Variables(n, a),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC))))

7dab87

\zeta\!\left(-n, 0\right) = -\frac{B_{n + 1}}{n + 1}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\zeta\!\left(-n, 0\right) = -\frac{B_{n + 1}}{n + 1}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`BernoulliB`	$B_{n}$	Bernoulli number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("7dab87"),
    Formula(Equal(HurwitzZeta(Neg(n), 0), Neg(Div(BernoulliB(Add(n, 1)), Add(n, 1))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

d99808

\zeta\!\left(0, a\right) = \frac{1}{2} - a

Assumptions:

a \in \mathbb{C}

TeX:

\zeta\!\left(0, a\right) = \frac{1}{2} - a

a \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d99808"),
    Formula(Equal(HurwitzZeta(0, a), Sub(Div(1, 2), a))),
    Variables(a),
    Assumptions(Element(a, CC)))

150b3e

\zeta\!\left(0, 0\right) = \frac{1}{2}

TeX:

\zeta\!\left(0, 0\right) = \frac{1}{2}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function

Source code for this entry:

Entry(ID("150b3e"),
    Formula(Equal(HurwitzZeta(0, 0), Div(1, 2))))

3db90c

\zeta\!\left(0, \frac{1}{2}\right) = 0

TeX:

\zeta\!\left(0, \frac{1}{2}\right) = 0

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function

Source code for this entry:

Entry(ID("3db90c"),
    Formula(Equal(HurwitzZeta(0, Div(1, 2)), 0)))

6419ac

\zeta\!\left(2, a\right) = {a}^{-2} \,{}_3F_2\!\left(1, a, a, a + 1, a + 1, 1\right)

Assumptions:

a \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\zeta\!\left(2, a\right) = {a}^{-2} \,{}_3F_2\!\left(1, a, a, a + 1, a + 1, 1\right)

a \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("6419ac"),
    Formula(Equal(HurwitzZeta(2, a), Mul(Pow(a, -2), Hypergeometric3F2(1, a, a, Add(a, 1), Add(a, 1), 1)))),
    Variables(a),
    Assumptions(Element(a, SetMinus(CC, ZZLessEqual(0)))))

448d90

\zeta\!\left(s, a\right) = \sum_{n=0}^{\infty} \frac{1}{{\left(n + a\right)}^{s}}

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\zeta\!\left(s, a\right) = \sum_{n=0}^{\infty} \frac{1}{{\left(n + a\right)}^{s}}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("448d90"),
    Formula(Equal(HurwitzZeta(s, a), Sum(Div(1, Pow(Add(n, a), s)), For(n, 0, Infinity)))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), Greater(Re(s), 1), Element(a, SetMinus(CC, ZZLessEqual(0))))))

77e507

\zeta^{(r)}\!\left(s, a\right) = {\left(-1\right)}^{r} \sum_{n=0}^{\infty} \frac{\log^{r}\!\left(n + a\right)}{{\left(n + a\right)}^{s}}

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

\zeta^{(r)}\!\left(s, a\right) = {\left(-1\right)}^{r} \sum_{n=0}^{\infty} \frac{\log^{r}\!\left(n + a\right)}{{\left(n + a\right)}^{s}}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

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
`Log`	$\log(z)$	Natural logarithm
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("77e507"),
    Formula(Equal(HurwitzZeta(s, a, r), Mul(Pow(-1, r), Sum(Div(Pow(Log(Add(n, a)), r), Pow(Add(n, a), s)), For(n, 0, Infinity))))),
    Variables(s, a, r),
    Assumptions(And(Element(s, CC), Greater(Re(s), 1), Element(a, SetMinus(CC, ZZLessEqual(0))), Element(r, ZZGreaterEqual(0)))))

0bd6aa

\zeta\!\left(s, N\right) = \sum_{n=N}^{\infty} \frac{1}{{n}^{s}}

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

TeX:

\zeta\!\left(s, N\right) = \sum_{n=N}^{\infty} \frac{1}{{n}^{s}}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`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("0bd6aa"),
    Formula(Equal(HurwitzZeta(s, N), Sum(Div(1, Pow(n, s)), For(n, N, Infinity)))),
    Variables(s, N),
    Assumptions(And(Element(s, CC), Greater(Re(s), 1), Element(N, ZZGreaterEqual(1)))))

60c6da

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

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

TeX:

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

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("60c6da"),
    Formula(Equal(HurwitzZeta(s, a), Add(Div(1, Sub(s, 1)), Sum(Mul(Mul(Div(Pow(-1, n), Factorial(n)), StieltjesGamma(n, a)), Pow(Sub(s, 1), n)), For(n, 0, Infinity))))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), Element(a, CC), NotElement(a, ZZLessEqual(0)))))

1699a9

\zeta\!\left(s, a\right) = \frac{\pi}{2 \left(s - 1\right)} \int_{-\infty}^{\infty} \frac{{\left(a - \frac{1}{2} + i x\right)}^{1 - s}}{\cosh^{2}\!\left(\pi x\right)} \, dx

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > \frac{1}{2}

References:

https://doi.org/10.1090/mcom/3401

TeX:

\zeta\!\left(s, a\right) = \frac{\pi}{2 \left(s - 1\right)} \int_{-\infty}^{\infty} \frac{{\left(a - \frac{1}{2} + i x\right)}^{1 - s}}{\cosh^{2}\!\left(\pi x\right)} \, dx

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > \frac{1}{2}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("1699a9"),
    Formula(Equal(HurwitzZeta(s, a), Mul(Div(Pi, Mul(2, Sub(s, 1))), Integral(Div(Pow(Add(Sub(a, Div(1, 2)), Mul(ConstI, x)), Sub(1, s)), Pow(Cosh(Mul(Pi, x)), 2)), For(x, Neg(Infinity), Infinity))))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), NotEqual(s, 1), Element(a, CC), Greater(Re(a), Div(1, 2)))),
    References("https://doi.org/10.1090/mcom/3401"))

498036

\zeta\!\left(s, a\right) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{{x}^{s - 1} {e}^{-a x}}{1 - {e}^{-x}} \, dx

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0

TeX:

\zeta\!\left(s, a\right) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{{x}^{s - 1} {e}^{-a x}}{1 - {e}^{-x}} \, dx

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Gamma`	$\Gamma(z)$	Gamma function
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("498036"),
    Formula(Equal(HurwitzZeta(s, a), Mul(Div(1, Gamma(s)), Integral(Div(Mul(Pow(x, Sub(s, 1)), Exp(Neg(Mul(a, x)))), Sub(1, Exp(Neg(x)))), For(x, 0, Infinity))))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), Greater(Re(s), 1), Element(a, CC), Greater(Re(a), 0))))

ed4f6f

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

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{or}}\; s = 0\right)

TeX:

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

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{or}}\; s = 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("ed4f6f"),
    Formula(Equal(HurwitzZeta(s, Add(a, 1)), Sub(HurwitzZeta(s, a), Div(1, Pow(a, s))))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), Element(a, CC), NotEqual(s, 1), Or(NotElement(a, ZZLessEqual(0)), Less(Re(s), 0), Equal(s, 0)))))

bed7ee

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

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{or}}\; s = 0\right) \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

TeX:

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

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{or}}\; s = 0\right) \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`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("bed7ee"),
    Formula(Equal(HurwitzZeta(s, Add(a, N)), Sub(HurwitzZeta(s, a), Sum(Div(1, Pow(Add(n, a), s)), For(n, 0, Sub(N, 1)))))),
    Variables(s, a, N),
    Assumptions(And(Element(s, CC), Element(a, CC), NotEqual(s, 1), Or(NotElement(a, ZZLessEqual(0)), Less(Re(s), 0), Equal(s, 0)), Element(N, ZZGreaterEqual(1)))))

95e270

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

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0\right) \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

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

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0\right) \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

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
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`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("95e270"),
    Formula(Equal(HurwitzZeta(s, Add(a, N), r), Add(HurwitzZeta(s, a, r), Mul(Pow(-1, Add(r, 1)), Sum(Div(Pow(Log(Add(a, k)), r), Pow(Add(a, k), s)), For(k, 0, Sub(N, 1))))))),
    Variables(s, a, N, r),
    Assumptions(And(Element(s, CC), Element(a, CC), NotEqual(s, 1), Or(NotElement(a, ZZLessEqual(0)), Less(Re(s), 0)), Element(N, ZZGreaterEqual(1)), Element(r, ZZGreaterEqual(0)))))

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)))))

ebc49c

\zeta\!\left(s, a\right) = \frac{1}{{2}^{s}} \left(\zeta\!\left(s, \frac{a}{2}\right) + \zeta\!\left(s, \frac{a + 1}{2}\right)\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

TeX:

\zeta\!\left(s, a\right) = \frac{1}{{2}^{s}} \left(\zeta\!\left(s, \frac{a}{2}\right) + \zeta\!\left(s, \frac{a + 1}{2}\right)\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

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("ebc49c"),
    Formula(Equal(HurwitzZeta(s, a), Mul(Div(1, Pow(2, s)), Add(HurwitzZeta(s, Div(a, 2)), HurwitzZeta(s, Div(Add(a, 1), 2)))))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), Element(a, CC), NotEqual(s, 1), Greater(Re(a), 0))))

7d9feb

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

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)))))

3ba544

\frac{d}{d s}\, \zeta\!\left(s, a\right) = \zeta'\!\left(s, a\right)

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0

TeX:

\frac{d}{d s}\, \zeta\!\left(s, a\right) = \zeta'\!\left(s, a\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("3ba544"),
    Formula(Equal(ComplexDerivative(HurwitzZeta(s, a), For(s, s)), HurwitzZeta(s, a, 1))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), NotEqual(s, 1), Element(a, CC), Greater(Re(a), 0))))

d0d03b

\frac{d^{r}}{{d s}^{r}} \zeta\!\left(s, a\right) = \zeta^{(r)}\!\left(s, a\right)

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

\frac{d^{r}}{{d s}^{r}} \zeta\!\left(s, a\right) = \zeta^{(r)}\!\left(s, a\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`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("d0d03b"),
    Formula(Equal(ComplexDerivative(HurwitzZeta(s, a), For(s, s, r)), HurwitzZeta(s, a, r))),
    Variables(s, a, r),
    Assumptions(And(Element(s, CC), NotEqual(s, 1), Element(a, CC), Greater(Re(a), 0), Element(r, ZZGreaterEqual(0)))))

83065e

\frac{d}{d a}\, \zeta\!\left(s, a\right) = -s \zeta\!\left(s + 1, a\right)

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \left\{0, 1\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0

TeX:

\frac{d}{d a}\, \zeta\!\left(s, a\right) = -s \zeta\!\left(s + 1, a\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \left\{0, 1\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("83065e"),
    Formula(Equal(ComplexDerivative(HurwitzZeta(s, a), For(a, a)), Neg(Mul(s, HurwitzZeta(Add(s, 1), a))))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), NotElement(s, Set(0, 1)), Element(a, CC), Greater(Re(a), 0))))

40c3e2

\frac{d^{r}}{{d a}^{r}} \zeta\!\left(s, a\right) = \left(1 - s - r\right)_{r} \zeta\!\left(s + r, a\right)

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; s + r \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

\frac{d^{r}}{{d a}^{r}} \zeta\!\left(s, a\right) = \left(1 - s - r\right)_{r} \zeta\!\left(s + r, a\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; s + r \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`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("40c3e2"),
    Formula(Equal(ComplexDerivative(HurwitzZeta(s, a), For(a, a, r)), Mul(RisingFactorial(Sub(Sub(1, s), r), r), HurwitzZeta(Add(s, r), a)))),
    Variables(s, a, r),
    Assumptions(And(Element(s, CC), NotEqual(s, 1), NotEqual(Add(s, r), 1), Element(a, CC), Greater(Re(a), 0), Element(r, ZZGreaterEqual(0)))))

d25d10

\zeta\!\left(s, a\right) = \sum_{k=0}^{N - 1} \frac{1}{{\left(a + k\right)}^{s}} + \frac{{\left(a + N\right)}^{1 - s}}{s - 1} + \frac{1}{{\left(a + N\right)}^{s}} \left(\frac{1}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \frac{\left(s\right)_{2 k - 1}}{{\left(a + N\right)}^{2 k - 1}}\right) - \int_{N}^{\infty} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} \frac{\left(s\right)_{2 M}}{{\left(a + t\right)}^{s + 2 M}} \, dt

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \operatorname{Re}\!\left(a + N\right) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}\!\left(s + 2 M - 1\right) > 0 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{or}}\; s = 0\right)

References:

http://dx.doi.org/10.1007/s11075-014-9893-1

TeX:

\zeta\!\left(s, a\right) = \sum_{k=0}^{N - 1} \frac{1}{{\left(a + k\right)}^{s}} + \frac{{\left(a + N\right)}^{1 - s}}{s - 1} + \frac{1}{{\left(a + N\right)}^{s}} \left(\frac{1}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \frac{\left(s\right)_{2 k - 1}}{{\left(a + N\right)}^{2 k - 1}}\right) - \int_{N}^{\infty} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} \frac{\left(s\right)_{2 M}}{{\left(a + t\right)}^{s + 2 M}} \, dt

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \operatorname{Re}\!\left(a + N\right) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}\!\left(s + 2 M - 1\right) > 0 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{or}}\; s = 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`BernoulliB`	$B_{n}$	Bernoulli number
`Factorial`	$n !$	Factorial
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Re`	$\operatorname{Re}(z)$	Real part
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("d25d10"),
    Formula(Equal(HurwitzZeta(s, a), Sub(Add(Add(Sum(Div(1, Pow(Add(a, k), s)), For(k, 0, Sub(N, 1))), Div(Pow(Add(a, N), Sub(1, s)), Sub(s, 1))), Mul(Div(1, Pow(Add(a, N), s)), Add(Div(1, 2), Sum(Mul(Div(BernoulliB(Mul(2, k)), Factorial(Mul(2, k))), Div(RisingFactorial(s, Sub(Mul(2, k), 1)), Pow(Add(a, N), Sub(Mul(2, k), 1)))), For(k, 1, M))))), Integral(Mul(Div(BernoulliPolynomial(Mul(2, M), Sub(t, Floor(t))), Factorial(Mul(2, M))), Div(RisingFactorial(s, Mul(2, M)), Pow(Add(a, t), Add(s, Mul(2, M))))), For(t, N, Infinity))))),
    Variables(s, a, N, M),
    Assumptions(And(Element(s, CC), NotEqual(s, 1), Element(a, CC), Element(N, ZZGreaterEqual(1)), Element(M, ZZGreaterEqual(1)), Greater(Re(Add(a, N)), 0), Greater(Re(Sub(Add(s, Mul(2, M)), 1)), 0), Or(NotElement(a, ZZLessEqual(0)), Less(Re(s), 0), Equal(s, 0)))),
    References("http://dx.doi.org/10.1007/s11075-014-9893-1"))

febdd2

\zeta\!\left(s\right) = \zeta\!\left(s, 1\right)

Assumptions:

s \in \mathbb{C}

TeX:

\zeta\!\left(s\right) = \zeta\!\left(s, 1\right)

s \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("febdd2"),
    Formula(Equal(RiemannZeta(s), HurwitzZeta(s, 1))),
    Variables(s),
    Assumptions(Element(s, CC)))

4228cd

B_{n}\!\left(z\right) = -n \zeta\!\left(1 - n, z\right)

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

B_{n}\!\left(z\right) = -n \zeta\!\left(1 - n, z\right)

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4228cd"),
    Formula(Equal(BernoulliPolynomial(n, z), Neg(Mul(n, HurwitzZeta(Sub(1, n), z))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(z, CC))))

53026a

\Gamma(z) = \sqrt{2 \pi} \exp\!\left(\zeta'\!\left(0, z\right)\right)

Assumptions:

z \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\Gamma(z) = \sqrt{2 \pi} \exp\!\left(\zeta'\!\left(0, z\right)\right)

z \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`Gamma`	$\Gamma(z)$	Gamma function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("53026a"),
    Formula(Equal(Gamma(z), Mul(Sqrt(Mul(2, Pi)), Exp(HurwitzZeta(0, z, 1))))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, ZZLessEqual(0)))))

f3b870

\log \Gamma(z) = \zeta'\!\left(0, z\right) + \frac{\log\!\left(2 \pi\right)}{2}

Assumptions:

z \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\log \Gamma(z) = \zeta'\!\left(0, z\right) + \frac{\log\!\left(2 \pi\right)}{2}

z \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("f3b870"),
    Formula(Equal(LogGamma(z), Add(HurwitzZeta(0, z, 1), Div(Log(Mul(2, Pi)), 2)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, ZZLessEqual(0)))))

693e0e

\psi\!\left(z\right) = \lim_{s \to 1} \left[\frac{1}{s - 1} - \zeta\!\left(s, z\right)\right]

Assumptions:

z \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\psi\!\left(z\right) = \lim_{s \to 1} \left[\frac{1}{s - 1} - \zeta\!\left(s, z\right)\right]

z \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`ComplexLimit`	$\lim_{z \to a} f(z)$	Limiting value, complex variable
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("693e0e"),
    Formula(Equal(DigammaFunction(z), ComplexLimit(Brackets(Sub(Div(1, Sub(s, 1)), HurwitzZeta(s, z))), For(s, 1)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, ZZLessEqual(0)))))

bba4ec

\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, z\right)

Assumptions:

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

TeX:

\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, z\right)

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("bba4ec"),
    Formula(Equal(DigammaFunction(z, m), Mul(Mul(Pow(-1, Add(m, 1)), Factorial(m)), HurwitzZeta(Add(m, 1), z)))),
    Variables(m, z),
    Assumptions(And(Element(m, ZZGreaterEqual(1)), Element(z, CC), NotElement(z, ZZLessEqual(0)))))

e05807

\log G(z) = \left(z - 1\right) \log \Gamma(z) - \zeta'\!\left(-1, z\right) + \zeta'(-1)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

TeX:

\log G(z) = \left(z - 1\right) \log \Gamma(z) - \zeta'\!\left(-1, z\right) + \zeta'(-1)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("e05807"),
    Formula(Equal(LogBarnesG(z), Add(Sub(Mul(Sub(z, 1), LogGamma(z)), HurwitzZeta(-1, z, 1)), ComplexDerivative(RiemannZeta(s), For(s, -1))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)))))

c31c10

L\!\left(s, \chi\right) = \frac{1}{{q}^{s}} \sum_{k=1}^{q} \chi(k) \zeta\!\left(s, \frac{k}{q}\right)

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \setminus \left\{1\right\}

TeX:

L\!\left(s, \chi\right) = \frac{1}{{q}^{s}} \sum_{k=1}^{q} \chi(k) \zeta\!\left(s, \frac{k}{q}\right)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \setminus \left\{1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("c31c10"),
    Formula(Equal(DirichletL(s, chi), Mul(Div(1, Pow(q, s)), Sum(Mul(chi(k), HurwitzZeta(s, Div(k, q))), For(k, 1, q))))),
    Variables(q, chi, s),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(s, SetMinus(CC, Set(1))))))

4c3678

\zeta\!\left(s, \frac{k}{q}\right) = \frac{{q}^{s}}{\varphi(q)} \sum_{\chi \in G_{q}} \overline{\chi(k)} L\!\left(s, \chi\right)

Assumptions:

q \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; k \in \{1, 2, \ldots, q - 1\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(k, q\right) = 1 \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \setminus \left\{1\right\}

TeX:

\zeta\!\left(s, \frac{k}{q}\right) = \frac{{q}^{s}}{\varphi(q)} \sum_{\chi \in G_{q}} \overline{\chi(k)} L\!\left(s, \chi\right)

q \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; k \in \{1, 2, \ldots, q - 1\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(k, q\right) = 1 \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \setminus \left\{1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`Totient`	$\varphi(n)$	Euler totient function
`Sum`	$\sum_{n} f(n)$	Sum
`Conjugate`	$\overline{z}$	Complex conjugate
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4c3678"),
    Formula(Equal(HurwitzZeta(s, Div(k, q)), Mul(Div(Pow(q, s), Totient(q)), Sum(Mul(Conjugate(chi(k)), DirichletL(s, chi)), ForElement(chi, DirichletGroup(q)))))),
    Variables(q, k, s),
    Assumptions(And(Element(q, ZZGreaterEqual(2)), Element(k, Range(1, Sub(q, 1))), Equal(GCD(k, q), 1), Element(s, SetMinus(CC, Set(1))))))

52ea5f

\operatorname{Li}_{s}\!\left(z\right) = \frac{\Gamma\!\left(1 - s\right)}{{\left(2 \pi\right)}^{1 - s}} \left({i}^{1 - s} \zeta\!\left(1 - s, \frac{1}{2} + \frac{\log\!\left(-z\right)}{2 \pi i}\right) + {i}^{s - 1} \zeta\!\left(1 - s, \frac{1}{2} - \frac{\log\!\left(-z\right)}{2 \pi i}\right)\right)

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left\{0, 1\right\} \;\mathbin{\operatorname{and}}\; s \notin \mathbb{Z}_{\ge 0}

TeX:

\operatorname{Li}_{s}\!\left(z\right) = \frac{\Gamma\!\left(1 - s\right)}{{\left(2 \pi\right)}^{1 - s}} \left({i}^{1 - s} \zeta\!\left(1 - s, \frac{1}{2} + \frac{\log\!\left(-z\right)}{2 \pi i}\right) + {i}^{s - 1} \zeta\!\left(1 - s, \frac{1}{2} - \frac{\log\!\left(-z\right)}{2 \pi i}\right)\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left\{0, 1\right\} \;\mathbin{\operatorname{and}}\; s \notin \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Gamma`	$\Gamma(z)$	Gamma function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("52ea5f"),
    Formula(Equal(PolyLog(s, z), Mul(Div(Gamma(Sub(1, s)), Pow(Mul(2, Pi), Sub(1, s))), Add(Mul(Pow(ConstI, Sub(1, s)), HurwitzZeta(Sub(1, s), Add(Div(1, 2), Div(Log(Neg(z)), Mul(Mul(2, Pi), ConstI))))), Mul(Pow(ConstI, Sub(s, 1)), HurwitzZeta(Sub(1, s), Sub(Div(1, 2), Div(Log(Neg(z)), Mul(Mul(2, Pi), ConstI))))))))),
    Variables(s, z),
    Assumptions(And(Element(s, CC), Element(z, CC), NotElement(z, Set(0, 1)), NotElement(s, ZZGreaterEqual(0)))))

Definitions

Illustrations

Domain and range

As a function of the argument

As a function of the parameter

Specific values

Series representations

Dirichlet series

Laurent series

Integral representations

Functional equations

Recurrence relations

Multiplication formula

Reflection formula

Derivatives and differential equations

Argument derivatives

Parameter derivatives

Euler-Maclaurin formula

Representation of other functions

Riemann zeta function

Bernoulli polynomials

Gamma and related functions

Dirichlet L-functions

Polylogarithms