Zeta function
From Exampleproblems
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
- Euler zeta function
- Hasse-Weil zeta function
- Hurwitz zeta function
- Ihara zeta function
- Igusa zeta function
- Lefschetz 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 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.
he:פונקצית זתה
