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Zeta FunctionThere 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: 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.
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