An Introduction to Zeta Functions

  • Pierre Cartier


In this Chapter, we aim at giving an elementary introduction to some functions which were found useful in number theory. The most famous is Riemann’s zeta function defined as follows
$$ \zeta \left( s \right) = \sum\limits_{n = 1}^\infty {n^{ - s} } = \prod\limits_p {\left( {1 - p^{ - s} } \right)^{ - 1} } $$
(where p runs over all prime numbers). This function provided the key towards a proof of the prime number distribution: as it was conjectured by Gauss and Legendre before 1830 and proved by Hadamard and de La Vallée Poussin in 1898, the number π(x) of primes p such that px is asymptotic to x/log x.


Functional Equation Prime Number Meromorphic Function Zeta Function Theta Function 
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A. General textbooks about number theory and works of historical value

  1. Blanchard, A. (1969) Introduction à la théorie analytique des nombres premiers, Dunod, 1969.Google Scholar
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B. Basic references for analytical methods

  1. Bellman, R. (1961) A brief introduction to theta functions, Holt, Rinehart and Winston, 1961.Google Scholar
  2. Bochner, S. (1932) Vorlesungen über Fouriersche Integrale, Leipzig, 1932.Google Scholar
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© Springer-Verlag Berlin Heidelberg 1992

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  • Pierre Cartier

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