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we maj introduce the Bernoui,lian functions fk{^), defined by tlie 

 equation 



- — r--^+ ^.^•VK6'j, 



and lience show that 



By equating tlie corresponding terms on the two sides we get 



:Ej\,n f-1 = (- 1)'"-' ^, ^ m(^)^^'--'" + ^ 



\n J Im: din 



as a first generalisation of the relation 



din 



Observing tliat we have 



there follows for two integers n and yz', both having the same set 

 of prime factors, 



^r f''^ 



\n 



In the same way an expression for the sum of the k^^ powers 

 of the integers v may be obtained. Expanding both sides of the 

 equation 



V djn e^<^ — 1 



we find 



^ :evI^ = 2 ii{d)dJ^Md!). 



k' J din 



Other relations of the same kind, containing trigonometrical functions 

 are deduced by changing x into 2jtii\ 

 From 



djn gSTTjarf — I 



we find by separating the real and imaginary parts 



2 cos 2jt,w z= i .s/n 2jr.vn S ft {d) cot Jt.vd^ 



dhi 



^ sin 27Ï.XV ■= sm* Ji.vn 2 ft (d) cot jxa'd. 



dju 



