A generalization of the Riemann Zeta Function with a Formula

(1) |

(2) |

(3) |

(4) |

(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) |

(10) | |

(11) | |

(12) | |

(13) |

**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.

© 1996-9

1999-05-25