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,...

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined...

zeta function of a variety Height zeta function of a variety Hurwitz zeta function, a generalization of the Riemann zeta function Igusa zeta function...

mathematics, the Lerch transcendent, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician...

(z)=\zeta _{H}'(0,z)-\zeta '(0),} where ζ H {\displaystyle \zeta _{H}} is the Hurwitz zeta function, ζ {\displaystyle \zeta } is the Riemann zeta function...

t\\&=(-1)^{m+1}m!\zeta (m+1,z)\end{aligned}}} where ζ ( s , q ) {\displaystyle \zeta (s,q)} is the Hurwitz zeta function. This expresses the polygamma function as the...

polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special...

} making it a special case of the Hurwitz zeta function ψ 1 ( z ) = ζ ( 2 , z ) . {\displaystyle \psi _{1}(z)=\zeta (2,z).} Note that the last two formulas...

periodic zeta function occurs in the reflection formula for the Hurwitz zeta function, which is why the relation that it obeys, and the Hurwitz zeta relation...

L-functions may be written as a linear combination of the Hurwitz zeta function at rational values. Fixing an integer k ≥ 1, the Dirichlet L-functions for...

coefficients of higher order with Gn(1) = Gn, Γ is the gamma function and ζ is the Hurwitz zeta function. Similar series with the Cauchy numbers of the second...

special functions and, in particular, the Riemann zeta function and the Hurwitz zeta function. They are an Appell sequence (i.e. a Sheffer sequence for...

first studied by Takuro Shintani (1976). They include Hurwitz zeta functions and Barnes zeta functions. Let P ( x ) {\displaystyle P(\mathbf {x} )} be a polynomial...

} The Legendre chi function appears as the discrete Fourier transform, with respect to the order ν, of the Hurwitz zeta function, and also of the Euler...

and the Riemann zeta function or the Hurwitz zeta function. Specifically, given a real number x, the rational zeta series for x is given by x = ∑ n = 2...

Dirichlet beta function Dirichlet L-function Hurwitz zeta function Legendre chi function Lerch transcendent Polylogarithm and related functions: Incomplete...

{1}{k}}-\log n-\sum _{m=2}^{\infty }{\frac {\zeta (m,n+1)}{m}}\right),} where ζ(s, k) is the Hurwitz zeta function. The sum in this equation involves the harmonic...

{\displaystyle N} approaches infinity, this becomes the Hurwitz zeta function ζ ( s , q ) {\displaystyle \zeta (s,q)} . For finite N {\displaystyle N} and q =...

quaternion Hurwitz scheme Hurwitz surface Hurwitz zeta function Hurwitz's automorphisms theorem Hurwitz's theorem (complex analysis) Hurwitz's theorem (composition...

Re(s) > 0. Alternatively, the following definition, in terms of the Hurwitz zeta function, is valid in the whole complex s-plane: β ( s ) = 4 − s ( ζ ( s...

The multiplication theorem for the Hurwitz zeta function ζ ( s , a ) = ∑ n = 0 ∞ ( n + a ) − s {\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }(n+a)^{-s}}...

{\bigl [}\zeta '(-1,z)-\zeta '(-1){\bigr ]}} where ζ′(z) denotes the derivative of the Riemann zeta function, ζ(a,z) denotes the Hurwitz zeta function and ζ...

{\displaystyle H_{n,m}=\zeta (m,1)-\zeta (m,n+1),} where ζ ( m , n ) {\displaystyle \zeta (m,n)} is the Hurwitz zeta function. This relationship is used...

tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function. The Clausen function of order 2 – often referred...

{\displaystyle -\partial _{s}\zeta _{H}(0,a)} , where ζ H ( s , a ) {\displaystyle \zeta _{H}(s,a)} is the Hurwitz zeta function. We will compute the determinant...

number Genocchi number Kummer's congruences Poly-Bernoulli number Hurwitz zeta function Euler summation Stirling polynomial Sums of powers Translation of...

polynomials are given in terms of the Hurwitz zeta function: ζ ( s , a ) = ∑ n = 0 ∞ 1 ( n + a ) s {\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }{\frac...

continuing ζ ( s ) {\displaystyle \zeta (s)} . Faulhaber's formula can be written in terms of the Hurwitz zeta function: ∑ k = 1 n k p = ζ ( − p ) − ζ (...

\left(k+q\right)^{s}}}~.} [clarification needed] The constant C is the Hurwitz zeta function evaluated at s. Zipfian distributions can be obtained from Pareto...

that paper, a slightly non-standard definition is used for the Hurwitz zeta function. Weisstein, Eric W. "Khinchin's constant". MathWorld. Ryll-Nardzewski...