Prof. Hardy, On Eamanujan's trigonometrical function, etc. 263 



Note on Ramanujans trigonometrical function Cq{n), and certain 

 series of arithmetical functions. By Prof. G. H. Hardy. 



^L [^Received 10 August 1920 : read 25 October.] 



1, Ramanujan's memoir ' On certain trigonometrical sums and 

 their applications in the theory of numbers'* is devoted to the 

 study of the function 



Cg(n) = ze =Scos — ^— (I'l). 



p V 9 



where p runs through the numbers less than and prime to q. 

 Ramanujan proves that 



c,(n) = Sg/.(|) (1-2), 



where fi(n) has its usual meaningf and S is a common divisor of 

 q and nl. He then proceeds to express a number of the most 

 important arithmetical functions of n as series of the form 



CO 



tagCg {7i), 



q = l 



where a^ is independent of ??. From among the many interesting 

 results which he obtains I may quote the following : — 



-«=?(^)^-°#* (i-^)' 



and in particular 



o-(«) = -g-s^-^ ^^'^^^' 



= t^~^ n.'^2); 



q 



.#,(«) = ^ 2 ''^l^^ (1-4); 



.W = -2t^P^ (1-5). 



25—1 



* Trans. Camh. Phil. Soc, vol. 22, 1918, pp. 259-276. 



t fM (»t) = unless m is a product of p different primes, when /j. (m) = ( - ly. 



"t The formula (1-2) seems to have been first stated explicitly by Ramanujan 

 (I.e. p. 260). It is given for n = 1 by Landau (Handbuc.li, 1909, p. 572) and by 

 Jensen ('Et nyt Udtryk for den talteoretiske Funktion 2// (») = 31 (m),' Saertryk 

 af Beretning om den 3 Skandinaviske Matematiker-Kongres, Kristiania, 1915). The 

 deduction of the general formula from that given by Landau is trivial; but 

 Ramanujan makes so many beautiful applications of the sums that they may well 

 be associated with his name. 



