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the power of ( - X ;i s), thus defining functions of s. A number of important theorems 

 on such series were proved by Dedekind, but both Dirichlet and Dedekind con- 

 sidered only real values of s. The first attempts to found a more general theory 

 of such series were made by Jensen in 1884 and 1888 and Cahen in 1894. In this 

 general theory, complex values of s were of course considered. The essential 

 difference which distinguishes the general theory of Dirichlet's series from the 

 simpler theory of power series is that, whereas the region of convergence of a 

 power series is determined simply by the disposition of the singular points of the 

 function which it represents, no such simple relation holds in the general case of 

 Dirichlet's series. Such a series, convergent in a portion only of the complex 

 plane, may represent a function regular all over the plane or in a region wider 

 than the region of convergence. In fact, roughly speaking, many of the peculiar 

 difficulties which attend the study of power series on the circle of convergence are 

 extended, in the case of Dirichlet's series, to wide regions of the plane or even to 

 the whole of it (pp. 9-10). 



This extremely able tract is, owing to the war, issued without Dr. Riesz's help 

 in the final correction of the proofs. This has given Mr. Hardy the opportunity 

 of mentioning in the preface the value of Dr. Riesz's contributions to the book 

 and the whole theory in general. Further, Mr. Hardy refers to his great debt 

 to the writings and personal encouragement of Dr. Edmond Landau, and this 

 partly explains the very touching dedication of this little volume. The subject of 

 Dirichlet's series and their applications to the theory of numbers up to 1909 has 

 been very fully dealt with by Landau in his Handbuch der Lehre von der Verteilung 

 der Primzahlen of the above date. The volume consists of eight chapters and a 

 very excellent bibliography of memoirs. This bibliography, though not professing 

 to be exhaustive, seems to be so as concerns recent literature on the subject. 

 After a short introduction (Chapter I.) we have a chapter (II.) on the elementary 

 theory of the convergence of Dirichlet's series. The region of convergence is a 

 half-plane, and the question of the convergence of the series on the line of conver- 

 gence remains open in general. The series also has a half-plane of absolute conver- 

 gence (pp 4, 8). The third chapter is on the formula for the sum of the coefficients 

 of a Dirichlet's series, and the order of the function represented by the series. Up 

 to this point only convergent Dirichlet's series have been considered ; such a series 

 defines an analytic function which may or may not exist outside the domain of 

 convergence of the series. In analogy with the methods of summation used of 

 oscillating series which have been used in the modern developments of the theory 

 of power series, we have in Chapter IV. an account of the summation of Dirichlet's 

 series by Riesz's "typical means," and in Chapter V. some general arithmetical 

 theorems concerning typical means. In Chapter VI. we have a treatment of 

 "Abelian and Tauberian theorems," and in Chapter VII. some further develop- 

 ments of the theory of functions represented by Dirichlet's series. Finally, 

 Chapter VIII. is on the multiplication of Dirichlet's series. 



Philip E. B. Jourdain. 



John Napier and the Invention of Logarithms, 1614. A lecture by E. W. 

 HOBSON, Sc.D., LL.D., F.R.S., Sadleirian Professor of Pure Mathematics, 

 Fellow of Christ's College, Cambridge. [Pp. 48.] (Cambridge : at the 

 University Press, 1914. Price is. 6d. net.) 



This lecture was delivered some time before the celebration at Edinburgh in 

 1914 of the tercentenary of the publication of John Napier's (1 550-161 7) Description 

 and this little book has a frontispiece reproduced from the steel engraving of 



