266 Prof. Hardy, Note on Ramanujan's trigonometrical function 



It may be immediately verified that these are also the values 

 of G^k{n). Hence Cq{n) = Gq{n) when g is a power of a prime, 

 and therefore generally. 



Summation of the series (1*3). 



5. It is plain that \Cg{n)\ does not exceed the sum of the 

 divisors of w. It therefore follows from (3-1) that, if s> 1, 



V Cg (n) 

 where y^ = 1 +M!^l) + ^^'(^)+... . 



If ^ + n, %^ = 1 - OT-*. 



If -st"- is the highest power of ot which divides n, 

 _ ^j-2 M^—l) t:7«-i(n7-l) 



A'w ~ -^ "• i:7~ "I z::r; 1- . . . + 



OT' 



OT«' cr 



(a+i)s 



= 1 -S3-«-(«+')«+ ot-«(ot- 1) = - 



1 — OT^"' 

 1 _ ^(a+i) (i-s) 



1 



•57 



.1— S 



TT ^ C„ (7}) 1 — ■ar(a+i) (i-s) 



Hence 2 -^ = H (1 - ^-^) H - — ~ - 



^ Oj_s (n) ^ w-(^-i) cr,_i (??.) 



This is Ramanujan's formula, with s in the place of s - 1. 



The formula (1-32) lies, as Ramanujan points out, somewhat 

 deeper, smce s = l does not lie inside the region of absolute con- 

 vergence. I have nothing to add to Ramanujan's remarks con- 

 cerning this limiting case*. 



Summation of the series (1*4). 

 6. Let us write, with Ramanujan, 



where p, p^, ... are the different prime factors of n, so that 

 <^i {n) = </) (01). Then, if s > 1, we have 



^ /^ (g) Cq (» ) TT 



where v - i + fli^h^i^ . 



9s (^) 



* I.e., p. 265. 



