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Hurwitz zeta function
Mathematics term

In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1, −2, … by

ζ ( s , a ) = ∑ n = 0 ∞ 1 ( n + a ) s . {\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }{\frac {1}{(n+a)^{s}}}.}

This series is absolutely convergent for the given values of s and a and can be extended to a meromorphic function defined for all s ≠ 1. The Riemann zeta function is ζ(s,1). The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882.

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Integral representation

The Hurwitz zeta function has an integral representation

ζ ( s , a ) = 1 Γ ( s ) ∫ 0 ∞ x s − 1 e − a x 1 − e − x d x {\displaystyle \zeta (s,a)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}e^{-ax}}{1-e^{-x}}}dx}

for Re ⁡ ( s ) > 1 {\displaystyle \operatorname {Re} (s)>1} and Re ⁡ ( a ) > 0. {\displaystyle \operatorname {Re} (a)>0.} (This integral can be viewed as a Mellin transform.) The formula can be obtained, roughly, by writing

ζ ( s , a ) Γ ( s ) = ∑ n = 0 ∞ 1 ( n + a ) s ∫ 0 ∞ x s e − x d x x = ∑ n = 0 ∞ ∫ 0 ∞ y s e − ( n + a ) y d y y {\displaystyle \zeta (s,a)\Gamma (s)=\sum _{n=0}^{\infty }{\frac {1}{(n+a)^{s}}}\int _{0}^{\infty }x^{s}e^{-x}{\frac {dx}{x}}=\sum _{n=0}^{\infty }\int _{0}^{\infty }y^{s}e^{-(n+a)y}{\frac {dy}{y}}}

and then interchanging the sum and integral.2

The integral representation above can be converted to a contour integral representation

ζ ( s , a ) = − Γ ( 1 − s ) 1 2 π i ∫ C ( − z ) s − 1 e − a z 1 − e − z d z {\displaystyle \zeta (s,a)=-\Gamma (1-s){\frac {1}{2\pi i}}\int _{C}{\frac {(-z)^{s-1}e^{-az}}{1-e^{-z}}}dz}

where C {\displaystyle C} is a Hankel contour counterclockwise around the positive real axis, and the principal branch is used for the complex exponentiation ( − z ) s − 1 {\displaystyle (-z)^{s-1}} . Unlike the previous integral, this integral is valid for all s, and indeed is an entire function of s.3

The contour integral representation provides an analytic continuation of ζ ( s , a ) {\displaystyle \zeta (s,a)} to all s ≠ 1 {\displaystyle s\neq 1} . At s = 1 {\displaystyle s=1} , it has a simple pole with residue 1 {\displaystyle 1} .4

Hurwitz's formula

The Hurwitz zeta function satisfies an identity which generalizes the functional equation of the Riemann zeta function:5

ζ ( 1 − s , a ) = Γ ( s ) ( 2 π ) s ( e − π i s / 2 ∑ n = 1 ∞ e 2 π i n a n s + e π i s / 2 ∑ n = 1 ∞ e − 2 π i n a n s ) , {\displaystyle \zeta (1-s,a)={\frac {\Gamma (s)}{(2\pi )^{s}}}\left(e^{-\pi is/2}\sum _{n=1}^{\infty }{\frac {e^{2\pi ina}}{n^{s}}}+e^{\pi is/2}\sum _{n=1}^{\infty }{\frac {e^{-2\pi ina}}{n^{s}}}\right),}

valid for Re(s) > 1 and 0 < a ≤ 1. The Riemann zeta functional equation is the special case a = 1:6

ζ ( 1 − s ) = 2 Γ ( s ) ( 2 π ) s cos ⁡ ( π s 2 ) ζ ( s ) {\displaystyle \zeta (1-s)={\frac {2\Gamma (s)}{(2\pi )^{s}}}\cos \left({\frac {\pi s}{2}}\right)\zeta (s)}

Hurwitz's formula can also be expressed as7

ζ ( s , a ) = 2 Γ ( 1 − s ) ( 2 π ) 1 − s ( sin ⁡ ( π s 2 ) ∑ n = 1 ∞ cos ⁡ ( 2 π n a ) n 1 − s + cos ⁡ ( π s 2 ) ∑ n = 1 ∞ sin ⁡ ( 2 π n a ) n 1 − s ) {\displaystyle \zeta (s,a)={\frac {2\Gamma (1-s)}{(2\pi )^{1-s}}}\left(\sin \left({\frac {\pi s}{2}}\right)\sum _{n=1}^{\infty }{\frac {\cos(2\pi na)}{n^{1-s}}}+\cos \left({\frac {\pi s}{2}}\right)\sum _{n=1}^{\infty }{\frac {\sin(2\pi na)}{n^{1-s}}}\right)}

(for Re(s) < 0 and 0 < a ≤ 1).

Hurwitz's formula has a variety of different proofs.8 One proof uses the contour integration representation along with the residue theorem.910 A second proof uses a theta function identity, or equivalently Poisson summation.11 These proofs are analogous to the two proofs of the functional equation for the Riemann zeta function in Riemann's 1859 paper. Another proof of the Hurwitz formula uses Euler–Maclaurin summation to express the Hurwitz zeta function as an integral

ζ ( s , a ) = s ∫ − a ∞ ⌊ x ⌋ − x + 1 2 ( x + a ) s + 1 d x {\displaystyle \zeta (s,a)=s\int _{-a}^{\infty }{\frac {\lfloor x\rfloor -x+{\frac {1}{2}}}{(x+a)^{s+1}}}dx}

(−1 < Re(s) < 0 and 0 < a ≤ 1) and then expanding the numerator as a Fourier series.12

Functional equation for rational a

When a is a rational number, Hurwitz's formula leads to the following functional equation: For integers 1 ≤ m ≤ n {\displaystyle 1\leq m\leq n} ,

ζ ( 1 − s , m n ) = 2 Γ ( s ) ( 2 π n ) s ∑ k = 1 n [ cos ⁡ ( π s 2 − 2 π k m n ) ζ ( s , k n ) ] {\displaystyle \zeta \left(1-s,{\frac {m}{n}}\right)={\frac {2\Gamma (s)}{(2\pi n)^{s}}}\sum _{k=1}^{n}\left[\cos \left({\frac {\pi s}{2}}-{\frac {2\pi km}{n}}\right)\;\zeta \left(s,{\frac {k}{n}}\right)\right]}

holds for all values of s.13

This functional equation can be written as another equivalent form:

ζ ( 1 − s , m n ) = Γ ( s ) ( 2 π n ) s ∑ k = 1 n [ e π i s 2 e − 2 π i k m n ζ ( s , k n ) + e − π i s 2 e 2 π i k m n ζ ( s , k n ) ] {\displaystyle \zeta \left(1-s,{\frac {m}{n}}\right)={\frac {\Gamma (s)}{(2\pi n)^{s}}}\sum _{k=1}^{n}\left[e^{\frac {\pi is}{2}}e^{-{\frac {2\pi ikm}{n}}}\zeta \left(s,{\frac {k}{n}}\right)+e^{-{\frac {\pi is}{2}}}e^{\frac {2\pi ikm}{n}}\zeta \left(s,{\frac {k}{n}}\right)\right]} .

Some finite sums

Closely related to the functional equation are the following finite sums, some of which may be evaluated in a closed form

∑ r = 1 m − 1 ζ ( s , r m ) cos ⁡ 2 π r k m = m Γ ( 1 − s ) ( 2 π m ) 1 − s sin ⁡ π s 2 ⋅ { ζ ( 1 − s , k m ) + ζ ( 1 − s , 1 − k m ) } − ζ ( s ) {\displaystyle \sum _{r=1}^{m-1}\zeta \left(s,{\frac {r}{m}}\right)\cos {\dfrac {2\pi rk}{m}}={\frac {m\Gamma (1-s)}{(2\pi m)^{1-s}}}\sin {\frac {\pi s}{2}}\cdot \left\{\zeta \left(1-s,{\frac {k}{m}}\right)+\zeta \left(1-s,1-{\frac {k}{m}}\right)\right\}-\zeta (s)} ∑ r = 1 m − 1 ζ ( s , r m ) sin ⁡ 2 π r k m = m Γ ( 1 − s ) ( 2 π m ) 1 − s cos ⁡ π s 2 ⋅ { ζ ( 1 − s , k m ) − ζ ( 1 − s , 1 − k m ) } {\displaystyle \sum _{r=1}^{m-1}\zeta \left(s,{\frac {r}{m}}\right)\sin {\dfrac {2\pi rk}{m}}={\frac {m\Gamma (1-s)}{(2\pi m)^{1-s}}}\cos {\frac {\pi s}{2}}\cdot \left\{\zeta \left(1-s,{\frac {k}{m}}\right)-\zeta \left(1-s,1-{\frac {k}{m}}\right)\right\}} ∑ r = 1 m − 1 ζ 2 ( s , r m ) = ( m 2 s − 1 − 1 ) ζ 2 ( s ) + 2 m Γ 2 ( 1 − s ) ( 2 π m ) 2 − 2 s ∑ l = 1 m − 1 { ζ ( 1 − s , l m ) − cos ⁡ π s ⋅ ζ ( 1 − s , 1 − l m ) } ζ ( 1 − s , l m ) {\displaystyle \sum _{r=1}^{m-1}\zeta ^{2}\left(s,{\frac {r}{m}}\right)={\big (}m^{2s-1}-1{\big )}\zeta ^{2}(s)+{\frac {2m\Gamma ^{2}(1-s)}{(2\pi m)^{2-2s}}}\sum _{l=1}^{m-1}\left\{\zeta \left(1-s,{\frac {l}{m}}\right)-\cos \pi s\cdot \zeta \left(1-s,1-{\frac {l}{m}}\right)\right\}\zeta \left(1-s,{\frac {l}{m}}\right)}

where m is positive integer greater than 2 and s is complex, see e.g. Appendix B in.14

Series representation

A convergent Newton series representation defined for (real) a > 0 and any complex s ≠ 1 was given by Helmut Hasse in 1930:15

ζ ( s , a ) = 1 s − 1 ∑ n = 0 ∞ 1 n + 1 ∑ k = 0 n ( − 1 ) k ( n k ) ( a + k ) 1 − s . {\displaystyle \zeta (s,a)={\frac {1}{s-1}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(a+k)^{1-s}.}

This series converges uniformly on compact subsets of the s-plane to an entire function. The inner sum may be understood to be the nth forward difference of a 1 − s {\displaystyle a^{1-s}} ; that is,

Δ n a 1 − s = ∑ k = 0 n ( − 1 ) n − k ( n k ) ( a + k ) 1 − s {\displaystyle \Delta ^{n}a^{1-s}=\sum _{k=0}^{n}(-1)^{n-k}{n \choose k}(a+k)^{1-s}}

where Δ is the forward difference operator. Thus, one may write:

ζ ( s , a ) = 1 s − 1 ∑ n = 0 ∞ ( − 1 ) n n + 1 Δ n a 1 − s = 1 s − 1 log ⁡ ( 1 + Δ ) Δ a 1 − s {\displaystyle {\begin{aligned}\zeta (s,a)&={\frac {1}{s-1}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n+1}}\Delta ^{n}a^{1-s}\\&={\frac {1}{s-1}}{\log(1+\Delta ) \over \Delta }a^{1-s}\end{aligned}}}

Taylor series

The partial derivative of the zeta in the second argument is a shift:

∂ ∂ a ζ ( s , a ) = − s ζ ( s + 1 , a ) . {\displaystyle {\frac {\partial }{\partial a}}\zeta (s,a)=-s\zeta (s+1,a).}

Thus, the Taylor series can be written as:

ζ ( s , x + y ) = ∑ k = 0 ∞ y k k ! ∂ k ∂ x k ζ ( s , x ) = ∑ k = 0 ∞ ( s + k − 1 s − 1 ) ( − y ) k ζ ( s + k , x ) . {\displaystyle \zeta (s,x+y)=\sum _{k=0}^{\infty }{\frac {y^{k}}{k!}}{\frac {\partial ^{k}}{\partial x^{k}}}\zeta (s,x)=\sum _{k=0}^{\infty }{s+k-1 \choose s-1}(-y)^{k}\zeta (s+k,x).}

Alternatively,

ζ ( s , q ) = 1 q s + ∑ n = 0 ∞ ( − q ) n ( s + n − 1 n ) ζ ( s + n ) , {\displaystyle \zeta (s,q)={\frac {1}{q^{s}}}+\sum _{n=0}^{\infty }(-q)^{n}{s+n-1 \choose n}\zeta (s+n),}

with | q | < 1 {\displaystyle |q|<1} .16

Closely related is the Stark–Keiper formula:

ζ ( s , N ) = ∑ k = 0 ∞ [ N + s − 1 k + 1 ] ( s + k − 1 s − 1 ) ( − 1 ) k ζ ( s + k , N ) {\displaystyle \zeta (s,N)=\sum _{k=0}^{\infty }\left[N+{\frac {s-1}{k+1}}\right]{s+k-1 \choose s-1}(-1)^{k}\zeta (s+k,N)}

which holds for integer N and arbitrary s. See also Faulhaber's formula for a similar relation on finite sums of powers of integers.

Laurent series

The Laurent series expansion can be used to define generalized Stieltjes constants that occur in the series

ζ ( s , a ) = 1 s − 1 + ∑ n = 0 ∞ ( − 1 ) n n ! γ n ( a ) ( s − 1 ) n . {\displaystyle \zeta (s,a)={\frac {1}{s-1}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\gamma _{n}(a)(s-1)^{n}.}

In particular, the constant term is given by

lim s → 1 [ ζ ( s , a ) − 1 s − 1 ] = γ 0 ( a ) = − Γ ′ ( a ) Γ ( a ) = − ψ ( a ) {\displaystyle \lim _{s\to 1}\left[\zeta (s,a)-{\frac {1}{s-1}}\right]=\gamma _{0}(a)={\frac {-\Gamma '(a)}{\Gamma (a)}}=-\psi (a)}

where Γ {\displaystyle \Gamma } is the gamma function and ψ = Γ ′ / Γ {\displaystyle \psi =\Gamma '/\Gamma } is the digamma function. As a special case, γ 0 ( 1 ) = − ψ ( 1 ) = γ 0 = γ {\displaystyle \gamma _{0}(1)=-\psi (1)=\gamma _{0}=\gamma } .

Discrete Fourier transform

The discrete Fourier transform of the Hurwitz zeta function with respect to the order s is the Legendre chi function.17

Particular values

Negative integers

The values of ζ(s, a) at s = 0, −1, −2, ... are related to the Bernoulli polynomials:18

ζ ( − n , a ) = − B n + 1 ( a ) n + 1 . {\displaystyle \zeta (-n,a)=-{\frac {B_{n+1}(a)}{n+1}}.}

For example, the n = 0 {\displaystyle n=0} case gives19

ζ ( 0 , a ) = 1 2 − a . {\displaystyle \zeta (0,a)={\frac {1}{2}}-a.}

s-derivative

The partial derivative with respect to s at s = 0 is related to the gamma function:

∂ ∂ s ζ ( s , a ) | s = 0 = log ⁡ Γ ( a ) − 1 2 log ⁡ ( 2 π ) {\displaystyle \left.{\frac {\partial }{\partial s}}\zeta (s,a)\right|_{s=0}=\log \Gamma (a)-{\frac {1}{2}}\log(2\pi )}

In particular, ζ ′ ( 0 ) = − 1 2 log ⁡ ( 2 π ) . {\textstyle \zeta '(0)=-{\frac {1}{2}}\log(2\pi ).} The formula is due to Lerch.2021

Relation to Jacobi theta function

If ϑ ( z , τ ) {\displaystyle \vartheta (z,\tau )} is the Jacobi theta function, then

∫ 0 ∞ [ ϑ ( z , i t ) − 1 ] t s / 2 d t t = π − ( 1 − s ) / 2 Γ ( 1 − s 2 ) [ ζ ( 1 − s , z ) + ζ ( 1 − s , 1 − z ) ] {\displaystyle \int _{0}^{\infty }\left[\vartheta (z,it)-1\right]t^{s/2}{\frac {dt}{t}}=\pi ^{-(1-s)/2}\Gamma \left({\frac {1-s}{2}}\right)\left[\zeta (1-s,z)+\zeta (1-s,1-z)\right]}

holds for ℜ s > 0 {\displaystyle \Re s>0} and z complex, but not an integer. For z=n an integer, this simplifies to

∫ 0 ∞ [ ϑ ( n , i t ) − 1 ] t s / 2 d t t = 2   π − ( 1 − s ) / 2   Γ ( 1 − s 2 ) ζ ( 1 − s ) = 2   π − s / 2   Γ ( s 2 ) ζ ( s ) . {\displaystyle \int _{0}^{\infty }\left[\vartheta (n,it)-1\right]t^{s/2}{\frac {dt}{t}}=2\ \pi ^{-(1-s)/2}\ \Gamma \left({\frac {1-s}{2}}\right)\zeta (1-s)=2\ \pi ^{-s/2}\ \Gamma \left({\frac {s}{2}}\right)\zeta (s).}

where ζ here is the Riemann zeta function. Note that this latter form is the functional equation for the Riemann zeta function, as originally given by Riemann. The distinction based on z being an integer or not accounts for the fact that the Jacobi theta function converges to the periodic delta function, or Dirac comb in z as t → 0 {\displaystyle t\rightarrow 0} .

Relation to Dirichlet L-functions

At rational arguments the Hurwitz zeta function may be expressed as a linear combination of Dirichlet L-functions and vice versa: The Hurwitz zeta function coincides with Riemann's zeta function ζ(s) when a = 1, when a = 1/2 it is equal to (2s−1)ζ(s),22 and if a = n/k with k > 2, (n,k) > 1 and 0 < n < k, then23

ζ ( s , n / k ) = k s φ ( k ) ∑ χ χ ¯ ( n ) L ( s , χ ) , {\displaystyle \zeta (s,n/k)={\frac {k^{s}}{\varphi (k)}}\sum _{\chi }{\overline {\chi }}(n)L(s,\chi ),}

the sum running over all Dirichlet characters mod k. In the opposite direction we have the linear combination24

L ( s , χ ) = 1 k s ∑ n = 1 k χ ( n ) ζ ( s , n k ) . {\displaystyle L(s,\chi )={\frac {1}{k^{s}}}\sum _{n=1}^{k}\chi (n)\;\zeta \left(s,{\frac {n}{k}}\right).}

There is also the multiplication theorem

k s ζ ( s ) = ∑ n = 1 k ζ ( s , n k ) , {\displaystyle k^{s}\zeta (s)=\sum _{n=1}^{k}\zeta \left(s,{\frac {n}{k}}\right),}

of which a useful generalization is the distribution relation25

∑ p = 0 q − 1 ζ ( s , a + p / q ) = q s ζ ( s , q a ) . {\displaystyle \sum _{p=0}^{q-1}\zeta (s,a+p/q)=q^{s}\,\zeta (s,qa).}

(This last form is valid whenever q a natural number and 1 − qa is not.)

Zeros

If a=1 the Hurwitz zeta function reduces to the Riemann zeta function itself; if a=1/2 it reduces to the Riemann zeta function multiplied by a simple function of the complex argument s (vide supra), leading in each case to the difficult study of the zeros of Riemann's zeta function. In particular, there will be no zeros with real part greater than or equal to 1. However, if 0<a<1 and a≠1/2, then there are zeros of Hurwitz's zeta function in the strip 1<Re(s)<1+ε for any positive real number ε. This was proved by Davenport and Heilbronn for rational or transcendental irrational a,26 and by Cassels for algebraic irrational a.2728

Rational values

The Hurwitz zeta function occurs in a number of striking identities at rational values.29 In particular, values in terms of the Euler polynomials E n ( x ) {\displaystyle E_{n}(x)} :

E 2 n − 1 ( p q ) = ( − 1 ) n 4 ( 2 n − 1 ) ! ( 2 π q ) 2 n ∑ k = 1 q ζ ( 2 n , 2 k − 1 2 q ) cos ⁡ ( 2 k − 1 ) π p q {\displaystyle E_{2n-1}\left({\frac {p}{q}}\right)=(-1)^{n}{\frac {4(2n-1)!}{(2\pi q)^{2n}}}\sum _{k=1}^{q}\zeta \left(2n,{\frac {2k-1}{2q}}\right)\cos {\frac {(2k-1)\pi p}{q}}}

and

E 2 n ( p q ) = ( − 1 ) n 4 ( 2 n ) ! ( 2 π q ) 2 n + 1 ∑ k = 1 q ζ ( 2 n + 1 , 2 k − 1 2 q ) sin ⁡ ( 2 k − 1 ) π p q {\displaystyle E_{2n}\left({\frac {p}{q}}\right)=(-1)^{n}{\frac {4(2n)!}{(2\pi q)^{2n+1}}}\sum _{k=1}^{q}\zeta \left(2n+1,{\frac {2k-1}{2q}}\right)\sin {\frac {(2k-1)\pi p}{q}}}

One also has

ζ ( s , 2 p − 1 2 q ) = 2 ( 2 q ) s − 1 ∑ k = 1 q [ C s ( k q ) cos ⁡ ( ( 2 p − 1 ) π k q ) + S s ( k q ) sin ⁡ ( ( 2 p − 1 ) π k q ) ] {\displaystyle \zeta \left(s,{\frac {2p-1}{2q}}\right)=2(2q)^{s-1}\sum _{k=1}^{q}\left[C_{s}\left({\frac {k}{q}}\right)\cos \left({\frac {(2p-1)\pi k}{q}}\right)+S_{s}\left({\frac {k}{q}}\right)\sin \left({\frac {(2p-1)\pi k}{q}}\right)\right]}

which holds for 1 ≤ p ≤ q {\displaystyle 1\leq p\leq q} . Here, the C ν ( x ) {\displaystyle C_{\nu }(x)} and S ν ( x ) {\displaystyle S_{\nu }(x)} are defined by means of the Legendre chi function χ ν {\displaystyle \chi _{\nu }} as

C ν ( x ) = Re χ ν ( e i x ) {\displaystyle C_{\nu }(x)=\operatorname {Re} \,\chi _{\nu }(e^{ix})}

and

S ν ( x ) = Im χ ν ( e i x ) . {\displaystyle S_{\nu }(x)=\operatorname {Im} \,\chi _{\nu }(e^{ix}).}

For integer values of ν, these may be expressed in terms of the Euler polynomials. These relations may be derived by employing the functional equation together with Hurwitz's formula, given above.

Applications

Hurwitz's zeta function occurs in a variety of disciplines. Most commonly, it occurs in number theory, where its theory is the deepest and most developed. However, it also occurs in the study of fractals and dynamical systems. In applied statistics, it occurs in Zipf's law and the Zipf–Mandelbrot law. In particle physics, it occurs in a formula by Julian Schwinger,30 giving an exact result for the pair production rate of a Dirac electron in a uniform electric field.

Special cases and generalizations

The Hurwitz zeta function with a positive integer m is related to the polygamma function:

ψ ( m ) ( z ) = ( − 1 ) m + 1 m ! ζ ( m + 1 , z )   . {\displaystyle \psi ^{(m)}(z)=(-1)^{m+1}m!\zeta (m+1,z)\ .}

The Barnes zeta function generalizes the Hurwitz zeta function.

The Lerch transcendent generalizes the Hurwitz zeta:

Φ ( z , s , q ) = ∑ k = 0 ∞ z k ( k + q ) s {\displaystyle \Phi (z,s,q)=\sum _{k=0}^{\infty }{\frac {z^{k}}{(k+q)^{s}}}}

and thus

ζ ( s , a ) = Φ ( 1 , s , a ) . {\displaystyle \zeta (s,a)=\Phi (1,s,a).\,}

Hypergeometric function

ζ ( s , a ) = a − s ⋅ s + 1 F s ( 1 , a 1 , a 2 , … a s ; a 1 + 1 , a 2 + 1 , … a s + 1 ; 1 ) {\displaystyle \zeta (s,a)=a^{-s}\cdot {}_{s+1}F_{s}(1,a_{1},a_{2},\ldots a_{s};a_{1}+1,a_{2}+1,\ldots a_{s}+1;1)} where a 1 = a 2 = … = a s = a  and  a ∉ N  and  s ∈ N + . {\displaystyle a_{1}=a_{2}=\ldots =a_{s}=a{\text{ and }}a\notin \mathbb {N} {\text{ and }}s\in \mathbb {N} ^{+}.}

Meijer G-function

ζ ( s , a ) = G s + 1 , s + 1 1 , s + 1 ( − 1 | 0 , 1 − a , … , 1 − a 0 , − a , … , − a ) s ∈ N + . {\displaystyle \zeta (s,a)=G\,_{s+1,\,s+1}^{\,1,\,s+1}\left(-1\;\left|\;{\begin{matrix}0,1-a,\ldots ,1-a\\0,-a,\ldots ,-a\end{matrix}}\right)\right.\qquad \qquad s\in \mathbb {N} ^{+}.}

Notes

References

  1. Hurwitz, Adolf (1882). "Einige Eigenschaften der Dirichlet'schen Functionen F ( s ) = ∑ ( D n ) ⋅ 1 n {\textstyle F(s)=\sum \left({\frac {D}{n}}\right)\cdot {\frac {1}{n}}} , die bei der Bestimmung der Classenanzahlen binärer quadratischer Formen auftreten". Zeitschrift für Mathematik und Physik (in German). 27: 86–101. /wiki/Adolf_Hurwitz ↩

  2. Apostol 1976, p. 251, Theorem 12.2 - Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 https://mathscinet.ams.org/mathscinet-getitem?mr=0434929 ↩

  3. Whittaker & Watson 1927, p. 266, Section 13.13 - Whittaker, E. T.; Watson, G. N. (1927). A Course Of Modern Analysis (4th ed.). Cambridge, UK: Cambridge University Press. ↩

  4. Apostol 1976, p. 255, Theorem 12.4 - Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 https://mathscinet.ams.org/mathscinet-getitem?mr=0434929 ↩

  5. Apostol 1976, p. 257, Theorem 12.6 - Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 https://mathscinet.ams.org/mathscinet-getitem?mr=0434929 ↩

  6. Apostol 1976, p. 259, Theorem 12.7 - Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 https://mathscinet.ams.org/mathscinet-getitem?mr=0434929 ↩

  7. Whittaker & Watson 1927, pp. 268–269, Section 13.15 - Whittaker, E. T.; Watson, G. N. (1927). A Course Of Modern Analysis (4th ed.). Cambridge, UK: Cambridge University Press. ↩

  8. See the references in Section 4 of: Kanemitsu, S.; Tanigawa, Y.; Tsukada, H.; Yoshimoto, M. (2007). "Contributions to the theory of the Hurwitz zeta-function". Hardy-Ramanujan Journal. 30: 31–55. doi:10.46298/hrj.2007.159. Zbl 1157.11036. https://doi.org/10.46298%2Fhrj.2007.159 ↩

  9. Apostol 1976, p. 257, Theorem 12.6 - Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 https://mathscinet.ams.org/mathscinet-getitem?mr=0434929 ↩

  10. Whittaker & Watson 1927, pp. 268–269, Section 13.15 - Whittaker, E. T.; Watson, G. N. (1927). A Course Of Modern Analysis (4th ed.). Cambridge, UK: Cambridge University Press. ↩

  11. Fine, N. J. (June 1951). "Note on the Hurwitz Zeta-Function". Proceedings of the American Mathematical Society. 2 (3): 361–364. doi:10.2307/2031757. JSTOR 2031757. Zbl 0043.07802. /wiki/Nathan_Fine ↩

  12. Berndt, Bruce C. (Winter 1972). "On the Hurwitz zeta-function". Rocky Mountain Journal of Mathematics. 2 (1): 151–158. doi:10.1216/RMJ-1972-2-1-151. Zbl 0229.10023. /wiki/Bruce_C._Berndt ↩

  13. Apostol 1976, p. 261, Theorem 12.8 - Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 https://mathscinet.ams.org/mathscinet-getitem?mr=0434929 ↩

  14. Blagouchine, I.V. (2014). "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations". Journal of Number Theory. 148. Elsevier: 537–592. arXiv:1401.3724. doi:10.1016/j.jnt.2014.08.009. /wiki/ArXiv_(identifier) ↩

  15. Hasse, Helmut (1930), "Ein Summierungsverfahren für die Riemannsche ζ-Reihe", Mathematische Zeitschrift, 32 (1): 458–464, doi:10.1007/BF01194645, JFM 56.0894.03, S2CID 120392534 https://eudml.org/doc/168238 ↩

  16. Vepstas, Linas (2007). "An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions". Numerical Algorithms. 47 (3): 211–252. arXiv:math/0702243. Bibcode:2008NuAlg..47..211V. doi:10.1007/s11075-007-9153-8. S2CID 15131811. /wiki/ArXiv_(identifier) ↩

  17. Jacek Klinowski, Djurdje Cvijović (1999). "Values of the Legendre chi and Hurwitz zeta functions at rational arguments". Mathematics of Computation. 68 (228): 1623–1631. Bibcode:1999MaCom..68.1623C. doi:10.1090/S0025-5718-99-01091-1. https://doi.org/10.1090%2FS0025-5718-99-01091-1 ↩

  18. Apostol 1976, p. 264, Theorem 12.13 - Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 https://mathscinet.ams.org/mathscinet-getitem?mr=0434929 ↩

  19. Apostol 1976, p. 268 - Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 https://mathscinet.ams.org/mathscinet-getitem?mr=0434929 ↩

  20. Berndt, Bruce C. (1985). "The Gamma Function and the Hurwitz Zeta-Function". The American Mathematical Monthly. 92 (2): 126–130. doi:10.2307/2322640. JSTOR 2322640. /wiki/Bruce_C._Berndt ↩

  21. Whittaker & Watson 1927, p. 271, Section 13.21 - Whittaker, E. T.; Watson, G. N. (1927). A Course Of Modern Analysis (4th ed.). Cambridge, UK: Cambridge University Press. ↩

  22. Davenport (1967) p.73 ↩

  23. Lowry, David (8 February 2013). "Hurwitz Zeta is a sum of Dirichlet L functions, and vice-versa". mixedmath. Retrieved 8 February 2013. http://mixedmath.wordpress.com/2013/02/08/hurwitz-zeta-is-a-sum-of-dirichlet-l-functions-and-vice-versa/ ↩

  24. Davenport (1967) p.73 ↩

  25. Kubert, Daniel S.; Lang, Serge (1981). Modular Units. Grundlehren der Mathematischen Wissenschaften. Vol. 244. Springer-Verlag. p. 13. ISBN 0-387-90517-0. Zbl 0492.12002. 0-387-90517-0 ↩

  26. Davenport, H. & Heilbronn, H. (1936), "On the zeros of certain Dirichlet series", Journal of the London Mathematical Society, 11 (3): 181–185, doi:10.1112/jlms/s1-11.3.181, Zbl 0014.21601 /wiki/Journal_of_the_London_Mathematical_Society ↩

  27. Davenport (1967) p.73 ↩

  28. Cassels, J. W. S. (1961), "Footnote to a note of Davenport and Heilbronn", Journal of the London Mathematical Society, 36 (1): 177–184, doi:10.1112/jlms/s1-36.1.177, Zbl 0097.03403 /wiki/Doi_(identifier) ↩

  29. Given by Cvijović, Djurdje & Klinowski, Jacek (1999), "Values of the Legendre chi and Hurwitz zeta functions at rational arguments", Mathematics of Computation, 68 (228): 1623–1630, Bibcode:1999MaCom..68.1623C, doi:10.1090/S0025-5718-99-01091-1 /wiki/Bibcode_(identifier) ↩

  30. Schwinger, J. (1951), "On gauge invariance and vacuum polarization", Physical Review, 82 (5): 664–679, Bibcode:1951PhRv...82..664S, doi:10.1103/PhysRev.82.664 /wiki/Physical_Review ↩