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:
Other zeta functions include:
- Artin-Mazur zeta-function
- Dedekind zeta-function
- Epstein 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.
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