120 My^ Mordell, On Mr Ramanujan's 



where a is a divisor of 12. Its expansion in powers of r involves 



only positive integral powers of r and starts with I — j r. 



/a(&)i, 6)2) is not however an invariant of the modular group. 

 We can avoid this difficulty by taking /(co,, eo,) = [A(ft)i, w^)]"'^'^. 

 In this case* 



Kp-ni 



^f(pco^,a),) =L(-1) ^ iyA(p&)i, &)2)J , 



provided we exclude p = 2 and p = S. Putting for the moment 



/m \" / "= —4-'! 



we find 



(—:) [iyA(a,„a,,)]- = S^^,ri-^^^ 





CO p-i OKpni ( a \ 2K7ri /■ «■ , „N 1 



2 



K = 



= 22 e"~6 



S = 0/<: = 



But since ^ 4" ^ or 3, p' —1=0 (mod 12). Hence 

 2 eP i 12 +*;=o, 



K = 



unless a{\ —p-)/l2 + s = O(mod_p), that is a 4- 12s = (mod jd), and 

 is then equal to p. Hence ^ is a power series in r^^^^ (really of the 

 form ?-'^/i2(^ + Br+ Gr" ...)), starting with r'<"'+'^^s)i\ip^ where s is 

 the smallest positive integer for which a + 12s = (mod p). Now 

 the only zeros of /(wi, oa^) = in the fundamental polygon are at 

 ft) = iX) or r = 0, and 



/(o)i, «2) = (^y '■'"' (1 + ^^' +^^ ...). 



But putting a = 1 2/6, so that h is an integer, 



a + ] 2s 1 + 6s 1 a 



12p bp "6^12' 



since 1 + 6s = (mod p). 



Hence <^lf{ui^, Wa) is a constant, and equations (8), (4), (5) 

 apply to the function 



V a 

 We note also that |= (- l)«(^-i)/2. 



* Hurwitz, I.e., p. 572, or Weber, Lelirhuch der Algebra, vol. 3, p. 252 



i 



«/12 



