Zeta function
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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
- Lerch zeta function
- Local 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 zeta functions should not be confused with the similar-sounding eta function.
he:פונקצית זתה