264 



surable. If the last two formulae l)e divided one hy the otiier, we 

 shall obtain, according to definition 6 



a-, (^i, n) «v (^,, 11) = «V {/f, n) . 



Proposition 4. If the following conditions are satisfied: 



k^ and k^ are two squareless incommensurable numbers and their 

 product is equal to k, 



n is an arbitrary integer and v an arbitrary number, pi'inie to h, 



D, is the G. C. D. of r— 1, k and k„ 



Z), is the G. C. D. of v — J, n and ^„ 



1) is the G. C. D. of v—\, n and k; 

 then we have 



Proof. D, aiul Z), are incommensurable, because /;, and k^ are 

 incommensurable. The numbers Z), and Z),, therefore also D^ Z),, 

 are divisors of D and Z) is a divisor of Z>, Z), ; consequently D.B^ = Z>. 

 Hence it follows 



li{V,) ix{D,) = (I (Z>), 



and 





A;, \ / ^ 



-Ix-l-zf ='^' "•/> 



according to 4, VIll. 



Proposition 5. If v be an integer, prime to the positive integer 

 k and the integers n and n' satisfy the relation rz/i' = 1 {mod. k), 

 then we shall have tv{n,k) ^=-y^-j{n' ,k). 



Proof. From the relation nn' = 1 [mod. k) it follows that n and 

 n' are prime to k and according to definition 5 and 4, III tAn,k) 

 and Xv(?i,^) are two conjugate functions with modulus J. 



Consequently 



Xv {ih ^)Xv (^» ^) = 1- 



Moreover we have, according to 4. I, II, and VI, 



Xv ("'. ^0 Xv {n, k) = Xv (nn', k) = /. (1, /t) = 1, 

 hence 



Xv(w, ^) — Xv("'- ^)' 

 Proposition 6. If ^ be a squareless number, prime to i' and D 

 represents the G. C. D. of r— 1, n and k, then we have 



k 



a,{n,k) = II {D) ff{D) t^ ( n, ^ 



