List of zeta functions

There are a number of mathematical functions with the name zeta-function, named after the Greek letter ζ. Of these, the most famous is the Riemann zeta-function.

Other zeta functions include:

• Artin-Mazur zeta-function
• Dedekind zeta-function
• Hasse-Weil zeta-function
• Hurwitz zeta-function
• Ihara zeta-function
• Igusa zeta-function
• Lefschetz zeta-function
• Lerch zeta-function
• Local zeta-function
• Minakshisundaram-Pleijel zeta function
• Selberg zeta-function
• Weierstrass zeta-function
• Zeta-function of a division algebra

Many of these zeta-functions are deeply related and are involved in a number of dramatic relationships. It is widely believed by mathematicians that there is a vast generalization that will tie much of the theory of zeta-functions and Dirichlet series together; but the nature of such a general theory is not known.

The Taniyama-Shimura theorem is one of the most recent advances towards that generalized understanding. Famous related conjectured relations include the Artin conjecture, the Birch and Swinnerton-Dyer conjecture and the generalized Riemann hypothesis. The theory of L-functions should in the end contain the theory of zeta-functions; an L-function is a potentially 'twisted' kind of zeta-function. The Selberg class S is an attempt to define zeta-functions axiomatically, so that the properties of the class can be studied, and the members of the class classified.

The zeta-functions should not be confused with the similar-sounding eta-function.