Introduction to Analytic Number Theory pp 224-248 | Cite as

# Dirichlet Series and Euler Products

Chapter

## Abstract

In 1737 Euler proved Euclid’s theorem on the existence of infinitely many primes by showing that the series ∑ for real where χ is a Dirichlet character and

*p*^{-1}, extended over all primes, diverges. He deduced this from the fact that the zeta function ζ(s), given by$$\zeta (s) = \sum\limits_{n = 1}^\infty {\frac{1}{{{n^s}}}}$$

(1)

*s*> 1, tends to ∞ as*s*→ 1. In 1837 Dirichlet proved his celebrated theorem on primes in arithmetical progressions by studying the series$$L(s,\chi ) = \sum\limits_{n = 1}^\infty {\tfrac{{\chi (n)}}{{{n^s}}}}$$

(2)

*s*> 1.## Keywords

Zeta Function Dirichlet Series Arithmetical Progression Multiplicative Function Riemann Zeta 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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© Springer Science+Business Media New York 1976