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

 It follows that 



f{s)= n |i + ^-T7*^ — rJ " (^--^3) 



n|i + 1 I n |(-^2)(^!:L^i)1 



= g{s)h{s), 

 say. Now 





= n 



1 + 



-^^)1 



OT«-l(t!7-l)J 



and this product is uniformly convergent throughout any half 

 plane o-^8> 0, so that g{s)/^{s) is regular for o-> 0. Thus g {s) 

 is regular for o- > 0, except for a simple pole at s = 1, with residue 

 1 ; and its properties for <t > are substantially the same as those 

 onis). 



If n is a prime p, or a power of a prime p, 



has a simple zero for s = 1, and 



Thus in this case/(s) is regular for s = 1, and 



/(l)=^log^- 



In all other cases (except when n=l), h{s) has a zero of at least 

 the second order, and 



/•(1) = 0. 



We have now only to apply to/(5) the arguments which, when 

 applied to the function l/^(s), lead to the proof of the well known 

 theorem expressed by the equation 



2^^ = 0* (7-2), 



n 



in order to conclude that the series for f{s) is convergent for 

 5 = 1, and that its value is /(I). That is to say 



2 ^'^^-V =- log 2^ 



<f>{q) p 



* See for example Landau, Handhiich, pp. 593 et seq. ; Hardy and Littlewood, 

 'New proofs of the prime-number theorem and similar theorems,' Quarterly 

 Journal, vol. 46, 1915, pp. 215-219. 



