( 413 ) 



d'—i , , 

 77 2 CO. — = 77 (- 1) 2 ' = (- l)if(«). 



If n =: '2m and //i be odd, we shall have r/ (//;) = 7; (?i). Half the 

 numbers y, prime to //i and less than m will be equal to some 

 integer v, the other half will be of the form v--in. 



Hence we have 



2jri? 2jrx , , , .cry. 



772 sin — = (— l)if(n) 77 2 sin = (— l)iK") 77 2 5<n — , 



V n y n X m 



and therefore 



77 2 Sin — 



ii sin / n \ 



/72co.- =(- 1)W.;^ ^ = (- l)W»).'UJ-''"> 



•J n _ Jiv 



77 2 sm — 

 n 



Ijastly, if n = 2m, and m be even, we shall have (f (ni) = è y ('O- 

 Now each of the numbers y. prime to m and less than rii at the 

 same time will be equal to some integer v and to one of the dif- 

 ferences V — ?M. Reasoning as before we have in this case 



2nrr , . / 2jry.y , . { ^ , ny\ 



772 mi = (— IjiK"; 77 2 sin = (— l)^''("i 77 2 sm — . 



n y \ 1^ J y. \ in J 



= (_ l)i-r(")e ^"^ 



From the foregoing we may conclude as follows. If we put 



77 2 cos '— = (— l)i?(« é''^«) , 



the arithmetical function >.(??) is different from zero only when 

 n is double the power of any prime number [>, in which case we 

 have X (n) =z lo(j jj . 



Again we introduce here the irreducible fractions Qn less than | 

 with the denominator ?z ; then denoting by ^J (q) the least common 

 multiple of all the integers not surpassing q we may write 



2 2 log 2 sin nr()„ = 2! Y{n) =^ log M{g), 



n < 7 " < .-7 



2 2 log 2 cos jtQn =: ^ ).(ti) =1 /o_a -1/ ( \ 



27* 



