Riemann Zeta Function - Generalizations

Generalizations

There are a number of related zeta functions that can be considered to be generalizations of the Riemann zeta function. These include the Hurwitz zeta function

(the convergent series representation was given by Helmut Hasse in 1930, cf. Hurwitz zeta function), which coincides with the Riemann zeta function when q = 1 (note that the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet L-functions and the Dedekind zeta-function. For other related functions see the articles Zeta function and L-function.

The polylogarithm is given by

which coincides with the Riemann zeta function when z = 1.

The Lerch transcendent is given by

which coincides with the Riemann zeta function when z = 1 and q = 1 (note that the lower limit of summation in the Lerch transcendent is 0, not 1).

The Clausen function Cl_s(θ) that can be chosen as the real or imaginary part of Li_s(e iθ).

The multiple zeta functions are defined by

One can analytically continue these functions to the n-dimensional complex space. The special values of these functions are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics.