312 



NATURE 



[November i8, 191 5 



DIRICHLET SERIES. 

 The General Theory of Dirichlet Series. By G. H. 

 Hardy and Marcel Riesz. Pp. x + 78. (Cam- 

 bridge: At the University Press, 1915.) Price 

 35. 6^. net. 



IT is well known that Dirichlet made a new 

 start in the theory of numbers by bringing it 

 into connection with certain analytical functions 

 of the form Sw/r"', f'h being an integer, and s a 

 real quantity. ' In the present tract the definition 

 of a Dirichlet series is 



y(j) = l^„exp (-X„t),. 



where (\„) is a sequence of real increasing numbers 

 converging to + 00, and s is a complex variable. 

 This obviously includes Dirichlet's functions, and 

 also those considered by Dedekind, Riemann, 

 Landau, and others, in the same connection. 



In the earlier chapters we have a summary of 

 the known elementary properties of f{s), such as 

 the conditions of its convergence when regarded 

 as a sum in the ordinary sense, and so on. But 

 the most novel, and principal, part, due mainly 

 to Dr. Riesz, is concerned with what he calls the 

 summability (\, k) of f{s). It is important that 

 the meaning of tliis should be properly under- 

 stood. Let Sn mean the sum of the first n terms 

 of f{s) ; then Dr. Riesz associates with this a func- 

 tion which we may call cr„ (A, k), where k is an arbi- 

 trary positive number, and A. is a functional sym- 

 bol defined by f{s) and k, so that we have a 

 sequence [<t„(\, /<)] which we may consider as 

 depending upon k. If this new sequence has a 

 limit L, we say that f{s) is summable (A, k). In 

 the definition we only consider the formal nature 

 of f{s), so that it does not matter whether /(.•?), 

 regarded as a series in the ordinary way, is con- 

 vergent or not. Thus we have a sort of extension 

 of the theories of Poincare, Borel, etc., about 

 divergent series. 



The conditions that L may exist give properties 

 of f{s) considered as a formal expression, and 

 hence certain conclusions of an arithmetical nature 

 can be drawn. It is in the further development 

 of these deductions that we may hope for further 

 information of interest. The mere fact that from 

 a sequence (5„) we may be able to construct a 

 sequence (o-„) which agrees, in the limit, with {$„) 

 when the latter is convergent, and may be called 

 its " sum " when it becomes divergent in the or- 

 dinary sense, is a barren definition until we apply 

 it to something concrete ; if we can do this, it 

 may be a very valuable resource, as in tVie case of 

 asymptotic summation. Everything goes to show 

 that the + we use in writing infinite series is in 

 some ways less appropriate than the comma of the 

 old-fashioned "progressions"; it will not matter 

 NO. 2403, VOL. 96] 



if we remember what we are doing, and that there 

 is no such thing as the actual sum of an infinite 

 number of terms. 



One observation in this tract is liable to mis- 

 understanding. It may be true that Cahen is the 

 first to discuss f{s) systematically as a function of 

 a complex variable 5 ; but the first step in this 

 direction was taken by Riemann in his famous 

 paper on the distribution of primes. To arith- 

 meticians, at any rate, it is the development of 

 Riemann 's results that is of principal interest at 

 present ; other problems of the same kind natur- 

 ally present themselves, as, for example, the 

 frequency of cases where p and (p + 2) are both 

 primes, like (5, 7) or (11, 13). The new analysis 

 may throw some light on these and other dark 

 places. In any case, this tract will be welcome 

 for its concise statement of known facts, and its 

 bibliography, which supplements that of Landau. 



G. B. M. 



OUR BOOKSHELF. 



Fungoid Diseases of Farm and Garden Crops. 

 By Dr. T. Mllburn. Pp. xi+ii8. (London: 

 Longmans, Green and Co., Ltd., 191 5.) Price 

 25. net. 



This book is intended " primarily for the use of 

 farmers, gardeners, and agricultural students," 

 and it is hoped that it may assist also "those 

 engaged in teaching and county lecturing." The 

 essentials of a book fulfilling these aims must be 

 that for the student the elementary scientific facts 

 are correctly and clearly stated, and for the farmer 

 and gardener that his understanding of the sub- 

 ject is developed by his attention being directed 

 to general principles while practical help Is given. 

 This book falls short of this standard. 



The farmer and gardener fail to get the know- 

 ledge they want when they are told that "no 

 exact formula " can be given for making Bor- 

 deaux mixture, and that "it is always necessary 

 to test the solution before applying, for if too 

 much lime be present it Is useless as a fungicide." 

 While the author quotes the titles of recent works 

 on the chemistry of Bordeaux mixture, it seems 

 scarcely possible that he has read them from the 

 account he gives of the chemical composition of 

 Bordeaux mixture. The statement that the pre- 

 paration of soda-Bordeaux Is "somewhat critical," 

 and that since this mixture "possesses no effectual 

 advantage " it can be passed over, will consider- 

 ably surprise the Irish Department of Agriculture 

 and the numberless farmers who, under their 

 tuition, have saved thousands of acres of potatoes 

 from " blight " by the u.se of this " soda-Bordeaux " 

 or "Burgundy mixture." The gardener and fruit- 

 grower will not acquiesce in the strange state- 

 ment that "washes of potassium sulphide" are 

 "very effectual, but too expensive." 



The book is well printed and very cheap; it 

 would have been better, however, to have in- 



