MATHEMATICS: G.'H. HARDY 
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e,(?) = l+^2 ( Sj ^)f (qe-™ /k ), (4) 
where 
'A, k 
and the summation applies to k = 1, 2, 3, . . . , and all positive values 
of h less than, of opposite parity to, and prime to k (h = 0 being associated 
with k = 1 alone). The coefficient of g w in Q s (q) is 
and our method leads to the conclusion that 
r s (n) = xs(n)+0(n h ), (6) 
at any rate for every value of s exceeding 4. 
When 5 is even, F(q) is an elementary function; and (Si, kY is easily expressi- 
ble in a form which does not involve the 'Legendre-Jacobi symbol' 
(;-> 
The function X s (n) is then susceptible of a variety of elementary transforma- 
tions and it was shown by Ramanujan, in the second of his two papers quoted 
above, that X s (n) is identical with r s (n) when s = 4, 6 or 8. In what follows 
I confine myself to the case in which s is odd, merely remarking that my method 
(which is entirely unlike that used by Ramanujan) leads directly to an alterna- 
tive proof of his results. 
3. When 5 is odd, F(q) is not an elementary function. But it is not diffi- 
cult to prove that 
F 
every term on the right hand side having an argument numerically less than 
Istt. Further, Si, k = S^l Sf,,k> and the first factor can always be expressed in 
a simple form. Suppose, to fix our ideas, that s = 5. Then Si,k = (— 1) A & 2 . 
Substituting from this equation and from (7) into (4), and effecting some ob- 
vious simplications, we obtain 
e, <*) = i + 2 ^tt^ nrhrnv (8) 
fif Vk [{h-kr)lY 
where now h assumes all values of opposite parity to and prime to k. This 
formula may be simplified further by multiplying each side by 
