Mr Mordell, On Mr Ramanujans Empirical Expansions, etc. 117 



On Mr Ravianujan's Empirical Expansions of Modular 

 Functions. By L. J. Mordell, Birkbeck College, London. (Com- 

 municated by Mr G. H. Hardy.) 



[Received 14 June 1917.] 



In his paper* "On Certain Arithmetical Functions" Mr 

 Ramanujan has found empirically some very interesting results 

 as to the expansions of functions which are practically modular 

 functions. Thus putting 



(^X^{<o„ CO,) = r [(1 - r) (1 - rO (1 - r^) . ..? = S T {n) r-, 



he finds that 



T{mn) = T(m)T{n) (1) 



if m and n are prime to each other ; and also that 



2 ^^ = Ul/(l-T(p)p-^+p^^-) (2), 



n=l "' 



where the product refers to the primes 2, 3, 5, 7 He also gives 



many other results similar to (2). 



My attention was directed to these results by Mr Hardy, and 

 I have found that results of this kind are a simple consequence 

 of the properties of modular functions. In the case above 



A (&)i, Wa) (r — e'"'^"", (o = coi/coj) 



is the well-known modular invariant (^f dimensions — 12 in co^, 0)2, 

 which is unaltered by the substitutions of the homogeneous 

 modular group defined by 



ft)/ = aoii + bco.2, (o./ = Cftji + dw2, 



where a, b, c, d are integers satisfying the condition ad —bc = l. 



Theorems such as T {mn) = T {m) T {n) had already been 

 investigated by Dr Glaisher f for other functions ; but the 

 theorems typified by equation (2) seem to be of a new type, and 

 it is very remarkable that they should have been discovered 

 empirically. The proof of Mr Ramanujan's formulae is as follows. 



Let f{(Oi, 6)2) be a modular^ form of dimensions ~ k in coj, co,, 

 which is a relative invariant of the homogeneous modular gi'oup, 

 so that /(fw/, w.^)l f{(o^, 0)2) is a constant independent of «i, (Oo,. 



* Transactions of the Cambridge Philosophical Society, vol. xxii. , no. ix. , 1916. 



t See, for example, his paper " The Arithmetical Functions P (m), Q (in), fi (;n) ", 

 Quarterlij Journal of Mathematics, vol. xxxvii., p. 36. 



+ For an elementary introduction to the modular functions, see Hurwitz, 

 Mathematische Annalen, vol. 18, p. 520, 



