242 Mr Shah and Mr Wilson, On an empirical formula connected 



so that (6) is equivalent to 



^ ^ (logw)^ j[9 — 2 g — 2 



Hence the asymptotic values furnished for v{n) by (6) and by (1) 

 are in the ratio 2e~'>' : 1, i.e. in the ratio 1123 : 1. 



A quite different formida was suggested by Stackel*, viz. 



(log?i)2 0(n) 

 where ^ (n) denotes, as usual, the number of numbers less than n 

 and prime to n. This is equivalent to 



V (n) ' 



P 



.(9). 



(log ny p — lq — 1 



Since p/(p—l) is nearer to unity than (p — l)/(p — 2), the 

 oscillations of v (n) would, if Stackel's formula were correct, be 

 decidedly less pronounced than they would be if (1) were correct. 

 As between the two formulae, the numerical evidence seems to be 

 decisive. Thus the ratio z^(8190) : z/(8192) is 3-32, whereas ac- 

 cording to (1) it should be 3"48, and according to Stackel's formula 

 it should be 2*37. Stackel's i-esult is obtained by considerations of 

 probability which ignore entirely the irregularity of the distribution 

 of the primes in a given interval ii^N, and it is not surprising, 

 therefore, that it should be seriously in error. 



On the other hand it should be observed that Sylvester's for- 

 mula (7) gives, within the range of the table on p. 240, very good 

 results, not much worse than those given by (5), and decidedly 

 better than those given by (1). This is shown by the table which 

 follows, in which decompositions into powers of primes higher than 

 the first are neglected. 



Gottinger Nachrichten (1896), pp. 292-299. 



