■I 



246 Mr Hardy and Mr Littleivood, Note on 



Now 

 (2-3) /(^) = 2A(?2)x"e2n/.'^^'9= S e=-^^^'^'"? 2 A(n)x>\ 



i = l n=,i 



If J is prime to q, we have* 

 (2-4) S A (n) X- = -^ 'S X. ( j) /. (X), 



where %« is the character conjugate to %«, and ^(5') is the number 

 of numbers less than and prime to q. It follows from (2'1) and 

 (2-2) that 



(2-5) S A (n) x» ^ 144 r^- = X?-M ^ • 



n=j </)(g) 1-X <l){q)l-K 



If on the other hand j is not prime to q, the formula (2'4) is 

 untrue, as its right-hand side is zero. But in this case A (??) = 

 unless n is a power of q, so that 



(2-6) S Ain)K^^=^o(~). 



From (2-3), (2-5), and (2-6) it follows that 



(2-7) •^^^•)~l~x' 



where 



^ ^ ' <P{q)7 ^{q)7' ' 



the summation extending over all values of j less than and prime 

 to q. The sum which appears in (2*71) has been evaluated by 

 Jensen and Ramanujanf, and its value is /Li{q), the well-known 

 arithmetical function of q which is equal to zero unless 5 is a product 

 JJ1P2 ■•• Pp of different primes, and then equal to (— 1)p. Thus 



(^•«> f^^>-mTh.t- 



3. The sum 

 (3-1) co{n)= t A(m)A(??0, 



■m + m'=n 

 * Landau, I.e., p. 421. 



m 



t J.L.W.V. Jensen, 'EtnytUdtryk for den talteoretiske Funktion 2M(n) = M{my, 



1 

 Saertryk af Beretning om den 3 Skandinaviske Matematiker-Kongres, Kristiania, 

 1915 ; S. Eamanu jan, ' On certain trigonometrical sums and their applications in the 

 theory of numbers', Trans. Camb. Phil. Soc, vol. 22, 1918, pp. 259-276. 



J If ^ (q) is zero, this formula is to be interpreted as meaning 



/w=o(r^) 



