Four Variants of Riemann Zeta Function

Abstract

By means of the generating function method and Dougall’s formulae for bilateral hypergeometric series, we examine four classes of infinite series, which may be considered as variants of Riemann zeta function. Several summation formulae are established in closed form, which shows remarkably that the values of these series result in multiples of integer powers of π either by rational numbers or by algebraic numbers.

Keywords

• Riemann zeta function
• the gamma function
• generating function
• trigonometric function
• bilateral hypergeometric series