## Hurwitz Zeta Function

A generalization of the Riemann Zeta Function with a Formula

 (1)

where any term with is excluded. The Hurwitz zeta function can also be given by the functional equation

 (2)

(Apostol 1976, Miller and Adamchik), or the integral

 (3)

If , then

 (4)

The Hurwitz zeta function satisfies
 (5) (6) (7)

where is the Gamma Function. The Polygamma Function can be expressed in terms of the Hurwitz zeta function by
 (8)

For Positive integers , , and ,

 (9)
where is a Bernoulli Number, a Bernoulli Polynomial, is a Polygamma Function, and is a Riemann Zeta Function (Miller and Adamchik). Miller and Adamchik also give the closed-form expressions

 (10) (11) (12) (13)

See also Khintchine's Constant, Polygamma Function, Psi Function, Riemann Zeta Function, Zeta Function

References

Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1995.

Elizalde, E.; Odintsov, A. D.; and Romeo, A. Zeta Regularization Techniques with Applications. River Edge, NJ: World Scientific, 1994.

Knopfmacher, J. Generalised Euler Constants.'' Proc. Edinburgh Math. Soc. 21, 25-32, 1978.

Magnus, W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. New York: Springer-Verlag, 1966.

Miller, J. and Adamchik, V. Derivatives of the Hurwitz Zeta Function for Rational Arguments.'' Submitted to J. Symb. Comput.

Spanier, J. and Oldham, K. B. The Hurwitz Function .'' Ch. 62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 653-664, 1987.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 268-269, 1950.