Messrs Shah and Wilsons paper 247 



which appears on the left-hand side of Shah and Wilson's equation 

 (2), is the coefficient of a;'* in the expansion of [/(«))". And 



when a; ^ e2i?7ri7(? along a radius vector. Our general method ac- 

 cordingly suggests to us to take 



n(n) = nt\^i(i^'e-'-^^P-il<i, 



where the summation extends over 5'= 1, 2, 3, ... and all values 

 of jt) less than and prime to q, as an approximation to co (n). Using 

 Ramanujan's notation, this sum may be written 



(3-2) n{n)=nl\^^l^%,(n). 



The series (3'2) can be summed in finite terms. We have 



(3'3) c,(7i) = SS/.(| 



the summation extending over all common divisors S of q and n*; 

 and it is easily verified, either by means of this formula or by means 

 of the definition of Cq{n) as a trigonometrical sum, that 



Cqq'{n) = Cq{n)Cg'(n) 



whenever q and q' are prime to one another. We may therefore 

 write 



n(n)=n:ZAq = nUx-r!r, 

 where the product extends over all primes -or, and 



since Aq contains the factor /j,(q) and A^^^., A ^-3, ... are accordingly 

 zero. 



If n is not divisible by zj, we have c^ (n) = /u, (ot) = — 1 and 



A =_^ 1 = L__. 



while if n is divisible by zr we have 



Cw('0 = At(t3-) + t3-/A(l)= OT - 1, 



. _ 1 



CT — 1 



Hence 



* Ramanujan, I.e., p. 260. 



