( 410 ) 



In particular the first of these equations gives a simple result if 



we put .^' = — + e, where e is a vanishing quantity. Asjhe factor 



dn 2 Jt,xn tends to zero with « the whole right-hand side is annulled 

 but for the term in which (/ = n. 

 So it follows that 



2 cos =: (I («), 



and we have fx (??), originally depending upon the prime factors of 

 n, expressed as a function of the integers prime to n. 



1 

 Similarly we may put in the second equation cc=: — and write 



sm nv ^ , ,. ^d 



^ = ^ fi ((i) cot —- . 



n din 2n 



Still another trigonometrical formula may be obtained by the sub- 

 stitution x = — -\- ^- Let D be the greatest common divisor of the 

 n 



integers n and q, so that 



n — n,D , q = q^D ; 



then as e vanishes, we have to retain at the right-hand side only 

 those terms in which qd is divisible by n, or what is the same the 

 terms for which the complementary divisor d' divides D. 

 Hence, Ave find 



:E cos ?^ = ^ ,x f 4 V' = ^ ^ f* ("o^) T • (*^' = ^> 



n d'lD \d J djD d 



Instead of extending the summation over all divisors d of D, it 



suffices to take into account only those divisors 6 of //, that are 



prime to ?ï„. In this way we find 



1 1 



D:E ii {n,d) — = fx (n„) D 2 (i (d) -- , 



dlD « 



and as the second side is readily reduced to 



<f{u) ( u ^ (f{n) 



n 



.A 



we obtain for any integer q, for which we have {n,q)^D, 







<i)' 



