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Originally published in 1915 as number eighteen in the Cambridge Tracts in Mathematics and Mathematical Physics series, and here reissued in its 1952 reprinted form, this book contains a condensed account of Dirichlet's Series, which relates to number theory. This tract will be of value to anyone with an interest in the history of mathematics or in the work of G. H. Hardy.
This classic work by two distinguished mathematicians explains theory and formulas behind Dirichlet's series and offers first systematic account of Riesz's theory of summation of series by typical means. 1915 edition.
This classic work by two distinguished mathematicians explains theory and formulas behind Dirichlet's series and offers first systematic account of Riesz's theory of summation of series by typical means. 1915 edition.
From the PREFACE. THE publication of this tract has been delayed by a variety of causes, and I am now compelled to issue it without Dr Riesz's help in the final correction of the proofs. This has at any rate, one advantage, that it gives me the opportunity of saying how conscious I am that whatever value it possesses is due mainly to his contributions to it, and in particular to the fact, that it contains the first systematic, account of his beautiful theory of the summation of series by 'typical means'. The task of condensing any account of so extensive a theory into the compass of one of these tracts has proved an exceedingly difficult one. Many important theorems are stated without proof, and many details are left to the reader. I believe, however, that our account is full enough to serve as a guide to other mathematicians researching in this and allied subjects. Such readers will be familiar with Landau's Handbuch der Lehre con der Verteilung der Primzahlen, and will hardly need to be told how much we, in common with all other investigators in this field, owe to the writings and to the personal encouragement of its author. G. H. H. * * * * * CONTENTS I. Introduction II. Elementary Theory of the Convergence of Dirichlet's Series III. The Formula for the Sum of the Coefficients of a Dirichlet's Series: The Order of the Function Represented by the Series IV. The Summation of Series by Typical Means V. General Arithmetic Theorems Concerning Typical Means VI. Abelian and Tauberian Theorems VII. Further Developments of the Theory of Functions Represented By Dirichlet's Series VIII. The Multiplication of Dirichlet's Series Bibliography
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A new edition of a classical treatment of elliptic and modular functions with some of their number-theoretic applications, this text offers an updated bibliography and an alternative treatment of the transformation formula for the Dedekind eta function. It covers many topics, such as Hecke’s theory of entire forms with multiplicative Fourier coefficients, and the last chapter recounts Bohr’s theory of equivalence of general Dirichlet series.
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