248 Mr Hardy and Mr Littlewood, Note on 



where 11' applies to primes which divide n and II" to primes 

 which do not. 



It is evident that O (n) is zero if n is odd. On the other hand, 

 if n is even, we have 



"W = 2»n|i-(-'-3y,}n 



-^)/|-.^-:. 



= 2^nn-^^ln^P-^ 



where tn- now runs through all odd primes and p through odd 

 prime divisors of n. 



The formula w (n) ^- il (n) 



is formula (2) of Shah and Wilson's paper*. 



The incorrectness of Sylvester's formula. 



4. It is easy to prove that if any form tda of the type 



(4-1) fw (7^) ~ CO (n) 



be true, then G must be unity. In other words, our formula is the 

 only formula of this type which can possibly be correct. This 

 may be shown as follows. 

 Let 



(4-2) f(^) = t'l^l 



where n runs through all even values; and let s — 1 = ^. The series 

 is absolutely convergent if s > 2, ^ > 1. Replacing 12 (n) by its 

 expression in terms of the prime divisors of n, and splitting up 

 f{s) into factors in the ordinary manner, we obtain 



say, where A is the same constant as in Shah and Wilson's paper, 

 and OT runs through all odd primes. 

 Let 



+ (*) = n (i + r^--.) = n [^}^_) = (1 - 2-0 f (0. 



and suppose that ^^1. Then 



x(0.nifi+^-4,*"4Vfi+i-^)} 



'y\r{t) [\ OT-2 1- 



•37 



V(^-2)| ^'1(37-1)^-11 A' 

 When fi(n) = 0, the formula is to be interpreted as meaning w(H) = o(/t). 



