## The General Theory of Dirichlet's SeriesThis classic work explains the theory and formulas behind Dirichlet's series and offers the first systematic account of Riesz's theory of the summation of series by typical means. Its authors rank among the most distinguished mathematicians of the twentieth century: G. H. Hardy is famous for his achievements in number theory and mathematical analysis, and Marcel Riesz's interests ranged from functional analysis to partial differential equations, mathematical physics, number theory, and algebra. Following an introduction, the authors proceed to a discussion of the elementary theory of the convergence of Dirichlet's series, followed by a look at the formula for the sum of the coefficients of a Dirichlet's series in terms of the order of the function represented by the series. They continue with an examination of the summation of series by typical means and of general arithmetic theorems concerning typical means. After a survey of Abelian and Tauberian theorems and of further developments of the theory of functions represented by Dirichlet's series, the text concludes with an exploration of the multiplication of Dirichlet's series. |

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### Contents

Introduction | 1 |

The formula for the sum of the coefficients of | 11 |

The summation of series by typical means | 19 |

Copyright | |

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### Common terms and phrases

Abel's theorem absolutely convergent Acta Math ambn apply argument arithmetic means Biesz Bohr Borel's Bromwich Cauchy's Theorem Cesaro's Chapman circle of convergence coefficient Comptes Rendus consider deduce defined Dirichletscher Reihen edition equation Felix Klein finite order follows from Theorem formula Fourier's series function G. H. Hardy Hardy and Littlewood Hence important theorem inequalities Infinite series infinity integral Knopp Landau Lemma limit Lindelof Lindelof's Theorem line of convergence logarithmic means Lond mean value theorem methods of summation obtain ordinary Dirichlet's series positive number positive values power series Proc proof of Theorem proves the theorem region of convergence regular Rendiconti di Palermo result Riesz Schnee second kind series is convergent series is summable series is uniformly suppose tends to zero term Theorem 17 Theorem 41 Theorems 23 theory of Dirichlet's tjber typical means Unabridged republication uniformly convergent uniformly summable vergence