Zeta function

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.

Related Topics:
Greek letter - ζ - Riemann zeta function

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Other zeta functions include:

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• Artin-Mazur zeta function
• Dedekind zeta function
• Euler 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.

Related Topics:
Vast generalization - Dirichlet series

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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.

Related Topics:
Taniyama-Shimura theorem - Artin conjecture - Birch and Swinnerton-Dyer conjecture - Generalized Riemann hypothesis - L-function - Selberg class S

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

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