2 - The Zeta and Related Functions

Abstract

This chapter provides a theory of the zeta and its related functions. It introduces the multiple Hurwitz zeta function ζn(s, a) and gives a detailed investigation of the properties and characteristics of this function. This function is a generalized form of the Riemann zeta function ζ(s). The Riemann zeta function ζ(s) is the most important member of the significantly large family of zeta functions The analytic continuation of ζn(s, a) is based on the convergence of an n-ple series. A special case of this function is the Hurwitz (or generalized) Zeta function ζ(s, a). For the case n = 1, ζn(s, a) becomes equal to ζ(s, a). The chapter also provides an overview of the polylogarithm functions, which include the Legendre's Chi function, Clausen's integral (or Clausen's function), and the Hurwitz–Lerch zeta function. These functions are useful in physics, electronics, quantum electrodynamics, and are related to other mathematical functions such as the Clausen integral or function.