270 Prof. Hardy, Note on Ramanujan's trigonometrical function 



If OT + 7i X-or = 1 - -^^^ ; 



'ST — 1 



while if ■CT* is the highest power of -nr which divides n 



X^ = 1 + t!7-^ + ^-'^ + . . . + 1!7-«^ - 



in- — 1 



Thus 



/(.)= n (i--^^) n (i + ^- + ... + .-«^-^r^" 



■w+n\ ^"J-Z-RTlnN -57 — 1 



= 9 («) ^' (*•)» 



say. It is plain that h (s) is a finite Dirichlet's series in s. On the 

 other hand, if 



,(.)?,(.)=^nj(i-^y(i_--)| 

 = n {1 + (zT-'^-') + {tiT-''^)} 



is regular and bounded in any half-plane <r ^ i + 5 > i. Thus g (s) 

 behaves, in any such half-plane, substantially like the reciprocal 

 of ^n(s)> or of ^(s); and so therefore does f{s). And so we can 

 prove (8-1) by substantially the same argument as that which 

 proves (7-2). 



Ii'or n = 1 we obtain 



<f>(q) ^^^)- 



It should be observed that, in this series, the'terms cancel in 

 pairs, since " 



/ii% ) ^ _ tM 



ifqis odd. Naturally this does not in itself establish the truth of 

 (8-2). 



Yj'r:^ ^^ ^^^ *^^* Ca(^) is equal to p,(q) if g is odd, to 

 -^(q) if q IS an odd multiple of 2, to 2/x(|g) if q is an odd 

 multiple of 4, and to zero if ^ = (mod 8). Thus we obtain 



l + _i___J: 2 1 1 ]__ 1 



</'(2) </>(3) 0(4) ct>{5) <p{6) 0(7) 0(10) 



1_ 2 



0(11)"^0(12) •••-^- 



