. The Clausen function can be chosen as the real or imaginary part of . The Lerch transcendent is given by which coincides with the Riemann zeta function when and (the lower limit of summation in the Lerch transcendent is 0, not 1). The multiple zeta functions are defined by One can analytically continue these functions to the -dimensional complex space. The special values taken by these functions at positive integer arguments are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics. See also 1 + 2 + 3 + 4 + ··· Arithmetic zeta function Generalized Riemann hypothesis Lehmer pair Prime zeta function Riemann Xi function Renormalization Riemann–Siegel theta function ZetaGrid Notes References Has an English translation of Riemann's paper. (Globally convergent series expression.) . In Gesammelte Werke, Teubner, Leipzig (1892), Reprinted by Dover, New York (1953). External links Riemann Zeta Function, in Wolfram Mathworld — an explanation with a more mathematical approach Tables of selected zeros Prime Numbers Get Hitched A general, non-technical description of the significance of the zeta function in relation to prime numbers. X-Ray of the Zeta Function Visually oriented investigation of where zeta is real or purely imaginary. Formulas and identities for the Riemann Zeta function functions.wolfram.com Riemann Zeta Function and Other Sums of Reciprocal Powers, section 23.2 of Abramowitz and Stegun Mellin transform and the functional equation of the Riemann Zeta function—Computational examples of Mellin transform methods involving the Riemann Zeta Function Visualizing the Riemann zeta function and analytic continuation a video from 3Blue1Brown Zeta and L-functions Analytic number theory Meromorphic functions Articles containing video clips Bernhard Riemann