Dirichlet Series and Euler Products

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


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


Zeta Function Dirichlet Series Arithmetical Progression Multiplicative Function Riemann Zeta Function 
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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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