Dirichlet Series and Euler Products

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.KeywordsCompact SubsetZeta FunctionIntegral FormulaDirichlet SeriesMultiplicative FunctionThese 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.