Hurwitz zeta function

In mathematics, the Hurwitz zeta function, named after Adolf Hurwitz, is one of the many zeta functions. It is formally defined for complex arguments s with Re(s) > 1 and q with Re(q) > 0 by

This series is absolutely convergent for the given values of s and q and can be extended to a meromorphic function defined for all s≠1. The Riemann zeta function is ζ(s,1).

Hurwitz zeta function corresponding to q = 1/3. It is generated as a Matplotlib plot using a version of the Domain coloring method.[1]

Analytic continuationEdit

 
Hurwitz zeta function corresponding to q = 24/25.

If   the Hurwitz zeta function can be defined by the equation

 

where the contour   is a loop around the negative real axis. This provides an analytic continuation of  .

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

 

where   is the gamma function and   is the digamma function.

Series representationEdit

 
Hurwitz zeta function as a function of q with s = 3+4i.

A convergent Newton series representation defined for (real) q > 0 and any complex s ≠ 1 was given by Helmut Hasse in 1930:[2]

 

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  ; that is,

 

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

 

Other series converging globally include these examples

 
 
 
 
 
 

where Hn are the Harmonic numbers,   are the Stirling numbers of the first kind,   is the Pochhammer symbol, Gn are the Gregory coefficients, G(k)
n
are the Gregory coefficients of higher order and Cn are the Cauchy numbers of the second kind (C1 = 1/2, C2 = 5/12, C3 = 3/8,...), see Blagouchine's paper.[3]

Integral representationEdit

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

 

for   and  

Hurwitz's formulaEdit

Hurwitz's formula is the theorem that

 

where

 

is a representation of the zeta that is valid for   and s > 1. Here,   is the polylogarithm.

Functional equationEdit

The functional equation relates values of the zeta on the left- and right-hand sides of the complex plane. For integers  ,

 

holds for all values of s.

Some finite sumsEdit

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

 
 
 

where m is positive integer greater than 2 and s is complex, see e.g. Appendix B in.[4]

Taylor seriesEdit

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

 

Thus, the Taylor series can be written as:

 

Alternatively,

 

with  .[5]

Closely related is the Stark–Keiper formula:

 

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 seriesEdit

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

 

Specifically   and  .

Fourier transformEdit

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

Relation to Bernoulli polynomialsEdit

The function   defined above generalizes the Bernoulli polynomials:

 

where   denotes the real part of z. Alternately,

 

In particular, the relation holds for   and one has

 

Relation to Jacobi theta functionEdit

If   is the Jacobi theta function, then

 

holds for   and z complex, but not an integer. For z=n an integer, this simplifies to

 

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  .

Relation to Dirichlet L-functionsEdit

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 (2s−1)ζ(s),[6] and if q = n/k with k > 2, (n,k) > 1 and 0 < n < k, then[7]

 

the sum running over all Dirichlet characters mod k. In the opposite direction we have the linear combination[6]

 

There is also the multiplication theorem

 

of which a useful generalization is the distribution relation[8]

 

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

ZerosEdit

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 1<Re(s)<1+ε for any positive real number ε. This was proved by Davenport and Heilbronn for rational or transcendental irrational q,[9] and by Cassels for algebraic irrational q.[6][10]

Rational valuesEdit

The Hurwitz zeta function occurs in a number of striking identities at rational values.[11] In particular, values in terms of the Euler polynomials  :

 

and

 

One also has

 

which holds for  . Here, the   and   are defined by means of the Legendre chi function   as

 

and

 

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.

ApplicationsEdit

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,[12] giving an exact result for the pair production rate of a Dirac electron in a uniform electric field.

Special cases and generalizationsEdit

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

 

For negative integer −n the values are related to the Bernoulli polynomials:[13]

 

The Barnes zeta function generalizes the Hurwitz zeta function.

The Lerch transcendent generalizes the Hurwitz zeta:

 

and thus

 

Hypergeometric function

  where  

Meijer G-function

 

NotesEdit

  1. ^ http://nbviewer.ipython.org/github/empet/Math/blob/master/DomainColoring.ipynb
  2. ^ Hasse, Helmut (1930), "Ein Summierungsverfahren für die Riemannsche ζ-Reihe", Mathematische Zeitschrift, 32 (1): 458–464, doi:10.1007/BF01194645, JFM 56.0894.03
  3. ^ Blagouchine, Iaroslav V. (2018). "Three Notes on Ser's and Hasse's Representations for the Zeta-functions". INTEGERS: The Electronic Journal of Combinatorial Number Theory. 18A: 1–45. arXiv:1606.02044. Bibcode:2016arXiv160602044B.
  4. ^ 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. Elsevier. 148: 537–592. arXiv:1401.3724. doi:10.1016/j.jnt.2014.08.009.
  5. ^ 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.
  6. ^ a b c Davenport (1967) p.73
  7. ^ Lowry, David. "Hurwitz Zeta is a sum of Dirichlet L functions, and vice-versa". mixedmath. Retrieved 8 February 2013.
  8. ^ Kubert, Daniel S.; Lang, Serge (1981). Modular Units. Grundlehren der Mathematischen Wissenschaften. 244. Springer-Verlag. p. 13. ISBN 0-387-90517-0. Zbl 0492.12002.
  9. ^ 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
  10. ^ 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
  11. ^ 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
  12. ^ 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
  13. ^ Apostol (1976) p.264

ReferencesEdit

External linksEdit