# Zeta function

Originally with zeta function or function in mathematics holomorphic complex function${\ displaystyle \ zeta}$ ${\ displaystyle \ zeta (z) = \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n ^ {z}}} \ quad}$ , With ${\ displaystyle z \ in \ mathbb {C}, \, \ Re (z)> 1}$ meant. Today this is called the Riemann zeta function , in honor of Bernhard Riemann , who around 1850 did important work to investigate this function in the complex. The study of the zeta function as a real function goes back to Leonhard Euler in the 1730s and 1740s, who among other things determined the values ​​of the zeta function for positive even-numbered arguments and found the product formula.

Some values ​​are

${\ displaystyle \ zeta (1) = \ infty}$ ${\ displaystyle \ zeta (2) = {\ frac {\ pi ^ {2}} {6}}}$ ${\ displaystyle \ zeta (3) = 1 {,} 2020569032 ...}$ ${\ displaystyle \ zeta (4) = {\ frac {\ pi ^ {4}} {90}}}$ ${\ displaystyle \ zeta (5) = 1 {,} 0369277551 ...}$ ${\ displaystyle \ zeta (6) = {\ frac {\ pi ^ {6}} {945}}}$ ${\ displaystyle \ zeta (7) = 1 {.} 0083492774 ...}$ ${\ displaystyle \ zeta (8) = {\ frac {\ pi ^ {8}} {9450}}}$ ${\ displaystyle \ zeta (9) = 1 {,} 0020083928 ...}$ Since then, many functions that are similar or generalizing in terms of definition or properties have been investigated, and the name Zeta function, along with that of its discoverer, was then given.

The most important other zeta functions are:

Also related to the Riemann zeta function, without having the “zeta” in the name, are the Dirichlet L series , Dirichlet eta function and Dirichlet beta function . ${\ displaystyle \ eta}$ ${\ displaystyle \ beta}$ ## literature

• Pierre Cartier : An introduction to Zeta Functions, in M. Waldschmidt u. a. (Ed.), From Number Theory to Physics, Springer 1992, pp. 1-63
• Anton Deitmar : A panorama of Zeta functions, in E. Kähler, Mathematical Works, De Gruyter 2003, Arxiv
• Mircea Mustaţă : Zeta functions in algebraic geometry, Lecture 2011 ( book.pdf pdf )
• Bernhard Schiekel: Zeta functions in physics - an introduction , doi : 10.18725 / OPARU-4418
• Alan David Thomas: Zeta-Functions: an introduction to algebraic geometry , Pitman 1977