Hurwitz zeta function

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In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for a complex argument s and a real argument q by

$\zeta(s,q) = \sum_{k=0}^\infty (k+q)^{-s}.$

This series is convergent for q > 0 and Re(s) > 1. If q is a non-positive integer it is assumed that terms in the series with denominator 0 be skipped. However, one usually confines oneself to 0 < q ≤ 1.

1 Analytic continuation
2 Series representation
3 Integral representation
4 Hurwitz's formula
5 Functional equation
6 Taylor series
7 Fourier transform
8 Relation to Bernoulli polynomials
9 Relation to Jacobi theta function
10 Relation to Dirichlet L-functions
11 Zeros
12 Rational values
13 Applications
14 Special cases and generalisations
15 References

The Hurwitz zeta function can be extended by analytic continuation to a meromorphic function defined for all complex numbers s with s ≠ 1. At s = 1 it has a simple pole with residue 1. The constant term is given by

$\lim_{s\to 1} \left[ \zeta (s,q) - \frac{1}{s-1}\right] = \frac{-\Gamma'(q)}{\Gamma(q)} = -\psi(q)$

where Γ is the Gamma function and ψ is the digamma function.

A convergent series representation defined for q > −1 and any complex s ≠ 1 was given by Helmut Hasse in 1930 ^[1]:

$\zeta(s,q)=\frac{1}{s-1} \sum_{n=0}^\infty \frac{1}{n+1} \sum_{k=0}^n (-1)^k {n \choose k} (q+k)^{1-s}.$

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 $q 1 - s$ ; that is,

$\Delta^n q^{1-s} = \sum_{k=0}^n (-1)^{n-k} {n \choose k} (q+k)^{1-s}$

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

$\zeta(s,q)=\frac{1}{s-1} \sum_{n=0}^\infty \frac{(-1)^n}{n+1} \Delta^n q^{1-s}$

$= \frac{1}{s-1} {\log(1 + \Delta) \over \Delta} q^{1-s}.$

The function has an integral representation in terms of the Mellin transform as

$\zeta(s,q)=\frac{1}{\Gamma(s)} \int_0^\infty \frac{t^{s-1}}{e^{qt}\left(1-e^{-t}\right)}dt$

for $\Re s>1$ and $\Re q >0$ .

Hurwitz's formula is the theorem that

$\zeta(1-s,x)=\frac{1}{2s}\left[e^{-i\pi s/2}\beta(x;s) + e^{i\pi s/2} \beta(1-x;s) \right]$

where

$\beta(x;s)= 2\Gamma(s+1)\sum_{n=1}^\infty \frac {\exp(2\pi inx) } {(2\pi n)^s}= \frac{2\Gamma(s+1)}{(2\pi)^s} \mbox{Li}_s (e^{2\pi ix})$

is a representation of the zeta that is valid for $0\le x\le 1$ and $s > 1$ . Here, $Li s (z)$ is the polylogarithm.

The functional equation relates values of the zeta on the left- and right-hand sides of the complex plane. For integers $1\leq m \leq n$ ,

$\zeta \left(1-s,\frac{m}{n} \right) = \frac{2\Gamma(s)}{ (2\pi n)^s } \sum_{k=1}^n \cos \left( \frac {\pi s} {2} -\frac {2\pi k m} {n} \right)\; \zeta \left( s,\frac {k}{n} \right)$

holds for all values of s.

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

$\frac {\partial} {\partial q} \zeta (s,q) = -s\zeta(s+1,q).$

Thus, the Taylor series has the distinctly umbral form:

$\zeta(s,x+y) = \sum_{k=0}^\infty \frac {y^k} {k!} \frac {\partial^k} {\partial x^k} \zeta (s,x) = \sum_{k=0}^\infty {s+k-1 \choose s-1} (-y)^k \zeta (s+k,x).$

Closely related is the Stark-Keiper formula:

$\zeta(s,N) = \sum_{k=0}^\infty \left[ N+\frac {s-1}{k+1}\right] {s+k-1 \choose s-1} (-1)^k \zeta (s+k,N)$

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

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

The function $β$ defined above generalizes the Bernoulli polynomials:

$B_n(x) = -\Re \left[ (-i)^n \beta(x;n) \right]$

where $\Re z$ denotes the real part of z. Alternately,

$\zeta(-n,x)=-{B_{n+1}(x) \over n+1}.$

In particular, the relation holds for $n = 0$ and one has

$\zeta(0,x)= \frac{1}{2} -x$

If $\vartheta (z,\tau)$ is the Jacobi theta function, then

$\int_0^\infty \left[\vartheta (z,it) -1 \right] t^{s/2} \frac{dt}{t}= \pi^{-(1-s)/2} \Gamma \left( \frac {1-s}{2} \right) \left[ \zeta(1-s,z) + \zeta(1-s,1-z) \right]$

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

$\int_0^\infty \left[\vartheta (n,it) -1 \right] t^{s/2} \frac{dt}{t}= 2\ \pi^{-(1-s)/2} \ \Gamma \left( \frac {1-s}{2} \right) \zeta(1-s) =2\ \pi^{-s/2} \ \Gamma \left( \frac {s}{2} \right) \zeta(s).$

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 Dirac delta function in z as $t\rightarrow 0$ .

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 q=1, when q=1/2 it is equal to (2^s-1)ζ(s), and if q=n/k with k>2, (n,k)>1 and 0<n<k, then

$\zeta(s,n/k)=\sum_\chi\overline{\chi}(n)L(s,\chi),$

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

$L(s,\chi)=\frac {1}{k^s} \sum_{n=1}^k \chi(n)\; \zeta \left(s,\frac{n}{k}\right).$

There is also the multiplication theorem

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

of which a useful generalization is

$\sum_{p=0}^{q-1}\zeta(s,a+p/q)=q^s\,\zeta(s,qa).$

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

If q=1 the Hurwitz zeta function reduces to the Riemann zeta function itself; if q=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<q<1 and q≠1/2, then there are zeros of Hurwitz's zeta function in the strip 1s)<1+ε for any positive real number ε. This was proved by Davenport and Heilbronn^[2] for rational and non-algebraic irrational q and by Cassels^[3] for algebraic irrational q.

The Hurwitz zeta function is engaged in a number of striking identities at rational values (given by Djurdje Cvijović and Jacek Klinowski, reference below). In particular, values in terms of the Euler polynomials $E n (x)$ :

$E_{2n-1}\left(\frac{p}{q}\right) = (-1)^n \frac{4(2n-1)!}{(2\pi q)^{2n}} \sum_{k=1}^q \zeta\left(2n,\frac{2k-1}{2q}\right) \cos \frac{(2k-1)\pi p}{q}$

and

$E_{2n}\left(\frac{p}{q}\right) = (-1)^n \frac{4(2n)!}{(2\pi q)^{2n+1}} \sum_{k=1}^q \zeta\left(2n+1,\frac{2k-1}{2q}\right) \sin \frac{(2k-1)\pi p}{q}$

One also has

$\zeta\left(s,\frac{2p-1}{2q}\right) = 2(2q)^{s-1} \sum_{k=1}^q \left[ C_s\left(\frac{k}{q}\right) \cos \left(\frac{(2p-1)\pi k}{q}\right) + S_s\left(\frac{k}{q}\right) \sin \left(\frac{(2p-1)\pi k}{q}\right) \right]$

which holds for $1\le p \le q$ . Here, the $C ν (x)$ and $S ν (x)$ are defined by means of the Legendre chi function $χ ν$ as

$C_\nu(x) = \operatorname{Re}\, \chi_\nu (e^{ix})$

and

$S_\nu(x) = \operatorname{Im}\, \chi_\nu (e^{ix}).$

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

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, given in 1951, giving an exact result for the pair production rate of a Dirac electron in a uniform electric field.

The Hurwitz zeta function generalizes the polygamma function:

ψ (m) (z) = ( - 1) m + 1 m!ζ(z, m + 1).

The Lerch transcendent generalizes the Hurwitz zeta:

$\Phi(z, s, q) = \sum_{k=0}^\infty \frac { z^k} {(k+q)^s}$

and thus

$\zeta (s,q)=\Phi(1, s, q).\,$

^ Helmut Hasse, Ein Summierungsverfahren fur die Riemannsche ζ-Reihe, (1930) Math. Z. 32 pp 458-464.
^ Davenport, H. and Heilbronn, H. On the zeros of certain Dirichlet series J. London Math. Soc. 11 (1936), pp. 181-185
^ Cassels, J. W. S. Footnote to a note of Davenport and Heilbronn J. London Math. Soc. 36 (1961), pp. 177-184

Jonathan Sondow and Eric W. Weisstein, Hurwitz Zeta Function at MathWorld.
Tom M. Apostol Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9 (See Chapter 12)
Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4. (See Paragraph 6.4.10 for relationship to polygamma function.)
Djurdje Cvijović and Jacek Klinowski, "Values of the Legendre chi and Hurwitz zeta functions at rational arguments", Mathematics of Computation 68 (1999), 1623-1630.
Victor S. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics, 100 (1998), pp 201--206.
Linas Vepstas, The Bernoulli Operator, the Gauss-Kuzmin-Wirsing Operator, and the Riemann Zeta

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