Empirical Expansions of Modular Functions 121 



When p = 2, these theorems hold if a = 4 or 12. For the 

 functions ^\f\* are selected as before, and it is clear that the 

 argument above applies, as « (1 —p^)l\2 is an integer. 



Lastly, when p = S, these theorems hold if a = 3, 6, 12, and the 

 functions ^k/k* are selected as before. 



Hence, altering our notation, we have the following theorems. 

 If a is a divisor of 12 and 



p 12 24 36 -j,,^ 00 



r [(l -r") (l-r«) (l -r") .-..J^" = S fa(n)r'\ 



n = \ 



then fa {m)fa (n) =fa (mn) (6), 



if m and n are prime to each other ; and 



^ />- (^) ^ n 1 /(i -^^ ^^^ I ^~ ^K^^''~' ] (7). 



n=i n' / V p' P~ 



The product refers to the primes 2, 3, 5, etc., except that 



p = 2 is excluded except when « = 4, 12, 

 and jt) = 3 is excluded except when a = .3, 6, 12. 



We notice that when a — 1, 2, 3, or 6, jt? = 2 is not excluded 

 as a factor of say m in (6), as in this case fai'ni) and fa (mn) are 

 both zero. Similarly for jo = 3 when a = 1, 2, 4. 



The result (6) is given by Mr Ramanujan wh6n a = 12, as are 

 most of the cases of (7). We shall now shew how in many cases 

 we can find simple expressions for fa{p). 



If a = 1, it is known that, by a result due to Kulerf, 



r[(l-?-i2)(l-r'^^)...]2 = [ 2 (-If ?• 2 J 



— 00 



(6ot + 1)2+(6w + 1)2 



= SS(-l)'"+"r 2 



= tt(-l)^r^'^^^\ 



where ^ = 3 (m + ?i) + 1, r) = n — m, so that ^, ij take all integer 

 values satisfying ^ = 1 (mod 3), ^ + 77 = 1 (mod 2). 



Hence f(p) — 2(— 1)'' if p = ^^ + 9r}^ and we take both ^ and 77 

 to be positive. If ^ = — 1 or ±5 (mod 12), f{p) is obviously zero. 

 This is Mr Ramanujan's result (118). 



If a = 2, it is known (Klein-Fricke, vol. 2, page 374) that 



r [(1 - r«) (1 - r'^) ...]* = ^2 (- 1)^ ^r^'+^^^+^r,^, 

 where f , r) take all integer values satisfying 



1=2 (mod 3), 7; = 1 (mod 2). 



* Hurwitz, I.e., vol. 18. 



t See also Klein-Fricke, Modulfunktionen, vol. 2, p. 374. 



VOL. XIX. PARTS II., III. 9 



