266 



{ki and k^ are two different prime numbers). According to I we have 



«., («, k,) = <Lt (I),) <p (D,) X. U ~\ 



D^ being the G. C. D. of v — 1, 7i and k^; 

 D, being the G. C. D. of v^\, n and A.,. 

 If tliese two formulae are multiplied, we shall obtain 



according to the piopositions 3 and 4. 



III. k is the product of three or four different prime factors. 



Take k equal to the product of the two incommensurable numbers 

 X-, and /t,, each of which is equal to a prime number or to the 

 product of two different [)riuie numbers. The proof is given in the 

 same way as in II. etc. 



Proposition 7. If k be a squareless number, prime to the integer 

 V and i>„4-; represents the G. C. I), of r — 1, n -\- ). and k, then 

 the in definition 3 defined coefficients Ay satisfy the relation 



k A, ,1 (A.-f >) </ (^A,4-;) Xv ( n H- /, --1- ) = 0, 



whatever be the value of the integers Ji and r. 

 Proof. 



27ij»(n 27ri/7)i 



9 



•/v {m, k) e f^ 2 A,e ^ =0, 



^=0 



for, if m and k have a common factor, /„ [m, k) = according to 



2 Trim 



4, III and if m is prime to k, e '^ is a primitive root of the equation 

 ri— 1 =0, i.e. a root of the equation 



Fk {x) = 1 A, .r> ^ 0, 

 ;,=o 



so that then the last factor is equal to zero. Hence 



27rmn 2mhn 



= ^ Xv (»». ^) «~^~ ^ A) e ^ 



, 2-nim {n-{-).) 



:= 2 A-^ S Xv (m, k) 



k 



/=0 ?n=l 



consequently 



2 A, ,1 (A,+ .) r/: (IA.+.) Xv ( n + ;, — ^T ^ Xv {m, k) e k 

 iz=o ' \ n 4- // ) f m=l 



according to definition 6 and proposition 6. 



