( 411 ) 



Concerning the result 



2;rr 



^' cos = (X (n) 



a slight remark may be made. To each integer v a second v' = n — v 

 is conjugated ; hence denoting by Qn an irreducible fraction <^ ^ with 

 the denominator n, we may write 



2^ cos 2jrQn = (i{n), 

 and also 



22 cos 2jrQn =■ 2 l^i^)- 



n _ 7 " — ."7 



Now for large values of </ the fractions Qn will spread themselves 

 not homogeneously, but still with some regularity more or less all 

 over the interval — ^ and there is some reason to expect, that in 

 the main the positive and the negative terms of the sum 2 cos 2jtQ,t 



uSg 



will annul each other, hence the equation 



22 cos 2jr()„ = 2 n{n) 



is quite consistent with the supposition of voN Sterneck, that as ^ takes 

 larger and larger values the absolute value of 2 n{n) does not 



exceed \/</. 



Another set of formulae will be obtained by substituting in 

 Kronecker's equation 



Ay) = log[e » — e 



71 ^ StiV 



2 log 2 sin — {v — x) = 2 (i{d) log 2 sin — . 



n djn ' d 



By repeated differentiations with respect to x we may derive from 

 this equation further analogies to the formula 



(p {n)r=z 2 n (d) d'. 



dhi 



So for instance we obtain bij diHerentiatiiig two times 



27 

 Proceedings Royal Acad. Amsterdam. Vol. IX. 



