244 Mr Shah and il/?- Wilson, On GoldbacJiS Theorem 



By an argument similar to that used in §4, in the reduction of 

 Sylvester's formula, it may be shown that this is equivalent to the 

 formula 



i.(n)~8^e-^y^ ,,^^...=4e-=yX(70 (12). 

 (log nf p-2 q- 2 



Thus this asymptotic value for v (n), and the Hardy-LittleAvood 

 value, are in the ratio 4e~-T : 1 = 1'263... : 1. Sylvester's is their 

 geometric mean. 



The formulae (11) and (12) would furnish a quite close ap- 

 proximation for V (n) for those values of oi on which it could be, in 

 practice, tested. Thus, for n = 170,170, we find that 



v{n)/4^e-'yX{n) = -9S.... 



But the ultimate incorrectness of the formula may be proved in 

 the same way as that of Sylvester's formula, namely by use of 

 Landau's asymptotic formula (10). 



Brun knew of the memoirs of Stackel and Landau, but appears 

 to have been unacquainted with Sylvester's work. 



