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Dirichlet Series and Euler Products

  • Tom M. Apostol
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

In 1737 Euler proved Euclid’s theorem on the existence of infinitely many primes by showing that the series Σ p − 1, extended over all primes, diverges. He deduced this from the fact that the zeta function ζ(s), given by
$$\zeta \left( s \right) = \sum\limits_{n = 1}^\infty {\frac{1}{{{n^s}}}} $$
(1)
for real s > 1, tends to 0o as s → 1. In 1837 Dirichlet proved his celebrated theorem on primes in arithmetical progressions by studying the series
$$L\left( {s,\chi } \right) = \sum\limits_{n = 1}^\infty {\frac{{\chi \left( n \right)}}{{{n^s}}}} $$
(2)
where χ is a Dirichlet character and s > 1.

Keywords

Compact Subset Zeta Function Integral Formula Dirichlet Series Multiplicative Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1976

Authors and Affiliations

  • Tom M. Apostol
    • 1
  1. 1.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA

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