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Mathematics. — ''Some formulae concerning the integers less than 

 n and prime to n." By Prof. J. C. Kluyver. 



The number ^ {ii) of tlie integers v less than n and prime to n 

 can be expressed by means of the divisors d. 

 We have 



(f (n) = -2" (1 (d) d\ [dd' = n) 



din 



if we denote b}' [i {q) the arithmetical function, whicli equals if q 

 be divisible by a square, and otherwise equals -|- 1 or — 1, according 

 to q being a product of an even or of an odd number of prime 

 numbers. 



This equation is a particular case of a more general one, by means 

 of which certain symmetrical functions of the integers v are expres- 

 sible as a function of the divisors (/. 



This general relation may be written as follows ^) 



k=d' 



2f{v) = 2ix{d)2f{kd). 



din l=\ 



For the proof we have to observe that, supposing (/??,, n) '^ D, the 

 term ƒ (w) occurs at the righthand side as often as d in a divisor 

 of D. Hence the total coefficient of the term f{m) becomes 



2 (x {d), 



dID 



that is zero if D be greater than unity, and J when m is equal to 

 one of the integers v. 



We will consider some simple cases of Kronecker's equation. 



First, let 



The equation becomes 



k=d' gxn 1 



^ e^-'= 2: (i{d)2 e^-^'i = 2 li (d.) e^(^ —. -, 



■J din i-—\ ,11» e^^ — 1 



or because of 



If Ave write 



din k=l djn 



2 H{d) z= 0, 



din 



gxn I 



din e^"^ — d 



:E — 2 aid) — , 



V e-n - 1 din e^'d - 1 



1) Kronecker, Vorlesungen fiber Zahlentheoiie. I, p. 251. 



