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



Here d (n) is the number of divisors of n, a (n) their sum, and 

 o-s(n) the sum of their s-th powers; (f){n) is the number of numbers 

 less than and prime to 71 ; 



when n~p"'pi'^'...; and r(n) is the number of representations of n 

 as the sum of two squares. 



These series have a peculiar interest because they show ex- 

 plicitly the source of the irregularities in the behaviour of their 

 sums. Thus, for example, the formula (1*31) may be written in the 

 form 



TT^/i f, (—1)" 2costw7r 2cosA??7r 



. („) = -g- |i + ^^ + -^^ + ^jf- 



2 (cos i?i7r + cos 4/i7r) 2cosin7r 

 H — ^^ 4- \- ... 



and we see at once that the most important term in a-{n) is ^tt^u, 

 and that irregular variations about this average value are produced 

 by a series of harmonic oscillations of decreasing amplitude. 



2. Ramanujan's proofs of his principal formulae are very 

 interesting and ingenious, but are not, I think, the simplest or 

 the most natural. In this note I prove a number of them by a 

 different method. This method occurred to Mr Littlewood and me 

 in the course of our researches connected with Waring's and Gold- 

 bach's problems, and in which Ramanujan's sums play an important 

 part. I also include a few new results which are suggested natur- 

 ally by our analysis. 



TJie multiplicative property ofCq(n). 



3. The first step is to prove that 



Cag'Oi) = Cg(n)Cgr(n) ...(3-1), 



whenever q and q are prime to one another. For this we observe 

 that 



Cg (n) Cg' (n) = S e-^'^P"'^'^ t e-2«i3''^^/a' = t e-2«-P'^i/3a' , 

 p p' P,p' 



where P = pq + p'q. 



But it is plain that every value of P is prime to both q and q , 

 and that no two values of P are congruent to modulus qq . Also 

 the total number of values of P is (^ {q) cf) (q') = (f) {qq')- Hence P 

 runs through a system of values congruent, to modulus qq, to the 

 cf) (qq) numbers less than and prime to qq'. This plainly establishes 

 the truth of (3-1). 



