Mr Hardy and Mr Littlewood, Note on Messrs Shah, etc. 245 



Note on Messrs Shah and Wilson's paper entitled: 'On an 

 empirical formula connected with Ooldbach's Theorem '. By G. H. 

 Hardy, M.A., Trinity College, and J. E. Littlewood, M.A., 

 Trinity College. 



[Received 22 January 1919: read 3 February 1919.] 



1. The formulae discussed by Messrs Shah and Wilson were 

 obtained in the course of a series of researches which have occupied 

 us at various times during the last two years. A full account of 

 our method will appear in due course elsewhere*: but it seems 

 worth while to give here some indication of the genesis of these 

 particular formulae, and others of the same character. We have 

 added a few words about various questions which are suggested by 

 Shah and Wilson's discussion. 



The genesis of the formulae. 



2. Let 



f{x) = %A (n) X'' = SA (n) e-"y = F{ij) 



and /, (x) = F, {y) = Ix^ (n) A (n) e"".", 



where A (n) is equal to logp when n is a prime p, or a power of ^j, 

 and to zero otherwise, and x< (^0 i^ one of Dirichlet's ' characters to 

 modulus (/'+. Also let 



X = xe'^'^''?, 



where p is positive, less than q, and prime to q ; and suppose that 

 X tends to unity by positive values. 

 It is known that 



n 



^X'^{v)A{v) = o{n\ 

 1 



unless Xk is the ' principal ' character Xi> ^^ which case 



n n 



1 1 



It follows that 



(2-1) A(K)'^-- ^ 



l-x 



and 



(2-2) f^^''^ = ^{l^ ^'^^l^- 



* An outline of one of its most important applications is contained in a paper 

 entitled ' A new solution of Waring's Problem ', which will be ijublished shortly in 

 the Quarterly Journal of Mathematics. 



t See Landau, Haudbuch, pp. 391 et seq. 



