ÖFVERSIGT AF K. VETENSK.-AKAD. FÖRHANDLINGAR 1901, N:0 9. 703 



where the series is supposed to be absolutely convergent for ?/ ^ 1. 

 A flrst condition for convergence is of course /(oo) = 0. 



If ds denotes the number of divisors in the integer s, tlie 

 number itself and unity being reckoned amongst theni, we im- 

 mediately find from (58) 



%F{sy)=%l4lsy). (59) 



s=l »—I 



Thus the series 



converges absolutely, because the series in (58) does so, and as 

 ds < 6^ the series 



and therefore also by (59) the series 



c» 



^F{sy) (60) 



is absolutely convergent. 



Now let (.i{s) denote MöBlus' coefficient corresponding to 

 the integer s, so that ^i{s) has the value + 1 for s = 1 and 

 the value ( — 1)*, when s =■ q^q^ . . . tjk, Qi, q2, ■■-, 'Jk being 

 unequal priiue numbers and the value O in all other cases, 

 which amounts to assume that s contains a quadratic factor. 



Converting both niembers of (58) we thus obtain 



2«(.)i^(.^)=/(y), (61) 



where the series converges absolutely and niore rapid than the 

 series (60). 



Let US now assume 



^(.!/) 





^..± («^2) 



