Zeta function

A zeta function is a function which is composed of an infinite sum of powers, that is, which may be written as a Dirichlet series:

:$zeta(s)\; =\; sum\_\{k=1\}^\{infty\}f(k)^s$

Examples

There are a number of mathematical functions with the name zeta-function, named after the Greek letter ζ; these zeta-functions should not be confused with the similar-sounding eta-functions.

Of the zeta-functions, the most famous is the Riemann zeta-function, for its involvement with the Riemann hypothesis, which is highly important in number theory.

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
* Prime zeta function
* Riemann zeta function
* Selberg zeta-function
* Weierstrass zeta-function

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 modularity 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.

A generalization for graphs and regular discrete lattices has been usedcite journal|author=O. Shanker|year=2007|title=Graph Zeta Function and Dimension of Complex Network|journal=Modern Physics Letters B|volume= 21(11)|pages=639–644|doi=10.1142/S0217984907013146] to give one possible definition of the dimension of a graph.

References

External links

* [http://www.maths.ex.ac.uk/~mwatkins/zeta/directoryofzetafunctions.htm A directory of all known zeta functions]