Bambah, R. P.

Chowla, S.

Gupta, H.

English

Publications of the IAS Fellows

A CONGRUENCE PROPERTY OF RAMANUJAN'S 
FUNCTION r(n) 
R. P. BAMBAH, S. CHOWLA, AND H. GUPTA 
We have recently proved1 that 
(1) r{n) s or(tt) (mod 8) 
if (n, 2) = 1, where 
00 00 
]T r(n)xn = * I I (1 - *n)24 ( I x I < !)i 
l l 
d\n 
We give, in this note, a new and short proof of (1). 
In the following we write J for any integral power series with in-
tegral coefficients. 
We have (l-x)2 = (l-x2) + 2J. Therefore 
(1 - s)8 = (l - *2)4 + 8/ . 
Hence 
00 
5>(«)«* = *{(1 - *2)(1 - *4)(1 - *6) • • • }12 + 8J 
i 
= x \ 53 ( - 1)*(2« + l)x"<w+1>l + 87 
v «=o ' 
(2) 
f °° } 4 
= « < £ *w(M+1) + 4/ > + 8 / 
But2 
Received by the editors September 11, 1946. 
1
 Jointly with D. B. Lahiri (Calcutta) in a paper sent to the Quarterly Journal of 
Mathematics, Oxford. 
2
 Whitaker and Watson, Modern analysis, 4th éd., p. 479, Example 3; and Rama-
nujan by Hardy, p. 132, for (l-f-2*+2*4+2x9-f • • • )* = 1+ X/4(w)xn and 
r4(w)=8cr°(w) where <r°(n)=<r(n) when n is odd and <70(n)=3 (the sum of the odd 
divisors of n) when n is even, that is, <r°(n) —<r(n) —4or(w/4) for all n. 
766 
RAMANUJAN'S FUNCTION T(«) 767 
16* { X) *" (u+1)} - 16*(1 + *2 + x* + a12 + • • • )4 
= (1 + 2x + 2x* + 2x' + • • • )« 
- (1 - 2* + 2** - 2»» -\ )4 
00 
(3) = 8 5 3 {*(*) - M»/4)}** 
l 
00 
- 8 2 3 {*(») - 4<r(»/4)}(- l)w*w 
l 
00 
= 1623 <r(n)xn (» is odd). 
n « l 
Therefore, from (2) and (3), we have 
00 00 
23 r(n)#w = 23 o-(»)*n + 8/ (n, 2) = 1. 
n = l n= l 
From the above, comparing coefficients, we have if (w, 2) = 1 
r(w) s flr(w) (mod 8) 
and if 21 w 
T(») S 0 (mod 8). 
THE UNIVERSITY OF THE PAN JAB, LAHORE, 
GOVERNMENT COLLEGE, LAHORE, AND 
GOVERNMENT COLLEGE, HOSHIARPUR 


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