238 Mr 8hah and Mr Wilson, On an empirical formula, connected 



On an empirical formula connected with GoldhacJis Theorem. 

 By N. M. Shah, Trinity College, and B. M. Wilson, Trinity Col- 

 lege. (Communicated by Mr G. H. Hardy.) 



[Received 20 January 1919 : read 3 February 1919.] 



§ 1. The following calculations originated in a request recently 

 made to us by Messrs G. H. Hardy and J. E. Littlewood, that we 

 should check a suggested asymptotic formula for the number of 

 ways V (n) of expressing a given even number n as the sum of two 

 primes. The formula in question is 



vin)^\{n) = 2Aj^^/^l^^^ (1), 



^ ^ ^ ^ (log ny p-2q — 2 ^ ^ 



where ??, = 2'^p«^^ ... (a^l) 



and A denotes the constant 



CO J- I 



p assuming, in this product, the odd prime values 3, 5. 7, 11, 13, .... 

 The formula (1) was deduced from another conjectured asymp- 

 totic formula, namely 



X A{m)A(m')^2An^^^ (2), 



where A (m) is the arithmetical function equal to log p when m is 

 a prime p, or a power oi p, and to zero otherwise, and the summation 

 on the left is extended to all pairs of positive integers m, m' such 

 that 



m + m = n. 



Formula (1) arises from (2) by replacing in the latter A{m) and 

 A {m) each by log n. It is natural, however, to expect a more 

 accurate result if we replace A (m) and A {ni) not by log n but by 

 \og^n, or, better still, if we replace the left-hand member of (2) by 



— ^ log X log {n — x)dx (3). 



1^ . 



The exact value of the expression (3) is found to be 



V (n) {(log ny - 2 log n + 2- i-tt^} (4). 



The various formulae thus obtained from (2) are, of course, all 

 asymptotically equivalent ; but the modified formulae are likely to 

 give more accurate results than (1) for comparatively small values 

 of n. We used the formula 



v(n)^p(n)=2A-. -^. ^^^~i (5), 



^ ^ • ^ ^ (log ny-2\ogn p — 2 q-2 ^ ^ 



obtained by ignoring the constant 2 — ^tt'^ in (4). 



