( 412 ) 



JtV 



sm' 



V 



d\n 



[X ((/) d! 



and bj repeating the process 



^2m 



— :e 



V L^^^ 



lm 



log sin y 



B,n 22'« 



j,2« 



.'/ = - 



^m d/n 



a result included in the still somewhat more a-eneral relation 



n^ 2 2 



V ]c=\ ynh—vf din 



which is self evident from. 

 Returning to the ec^uation 



T€ . JttV 



2 log 2 dn y — [v — x) z=z ^ n [d) log 2 sin — 

 V " n din d 



we obtain as x tends to zero 



2 log 2 sin — = — 2 [i {d) log d. 



V '»' din 



111 order to evaluate the right-hand side, we observe that for 

 n = Pi"-^ p/'- • . . we have 



d 



S f* (d) log d = 



dlii 



dy 



(1 — ell H Pi) (1 — eV^oöiJ'j) . 



,V=rU 



So it is seen that, putting 



— 2 n (d) log d = Y (n), 



din 



the function y {n) is equal to zero for all integers n having distinct 

 prime factors, and that it takes the value loc/ p, when n is any powei 

 of the prime number p. 

 Hence we may write 



JtV 



112 sin — = ev W, 

 V n 



a result in a different way deduced by Kronecker ^). 

 Again in the equation 



n 2 sin — (r — .v) = /ƒ I 2 sin 



din 



d 



we will make ,v tend to — 



If n be odd, all divisors (/ and (/' are odd also and we have 

 at once 



^) Kronecker, Yorlesungen über Zahlentheorie. I, p. 296, 



