263 



VIL :i^ Xv (n, k) = o, if V ~\^ 1 {mod. k). 



u—i 



VIII. Xv (", ^-J Xv (n, /:,) = Xv («, fc), 



if /• be equal to the prodncl of the two incommensurable integers 



k^ and /i-,. 



Definition 5. -/j^{n,k) and X;{n,k} are two conjugate functions; 

 they are, therefore, identical, if x.,{n,k) is real and they are conjugate 

 imaginary, if yi;{n,k) is an imaginary function. 



Proposition 2. y,;{iij>-) is an arithmetical character of 7i, modulo ^'. 



Proof. Each function of n, satisfying the conditions 4.1, 11 and III, 

 is an ai-ithmetical character of Ji, modulo k. 



Definition 6. The functions a.,{n,k) of the variable integer ii, 

 (r and X' being incommensurable numbers) are defined by the relation 



, 2Tr imn , Stt im 



2 Xv (^^ k}e ^ = a, (n, k) ^ Xv (»N -^O ^ * ■ 



»i = 1 ;yi = 1 



Proposition 3. If /• be the product of the two incommensurable 

 numbers l\ and k„ each of which is prime to the integer v, then 

 we shall have 



ttv (n, ^j) av (n, ^,) =: a,, (?;, /:). 



Proof. In the expression 



mi=l (/»2=-.l )«i=^l ;/i-=l 



we have ???, = 1, 2, 3, . . . Z', and ?/?.^ := J , 2, 3, . . . , /•,. We rtiay make 

 rit^k^ -\- in^k-^ congruent to m {mod. k) and k'> rii'^1. Then we 

 have m ::= I, 2, 3, . . . ^' and 



Xv (w, k) = Vv (m, k^) Xv (?», ^,) (according to 4, VIII) 



=z i.,{m^k^, kj)y;{m^k^, kj) (according to 4, II) 



= Xv(^,»^i)Xv(^'i» ^,) Xv(m,, ^J Xv("t,. ^*,) (according to 4, I). 

 Consequently 



, 2Ttiunn T 2711111.^11 , 2:r!mn 



Xv {k,, ^-J Xv f/^-r ^-J ^ Xv (m,,/;,)g~^:S' Xv (m,>,)g "^^ = 25' x, (m, /t) e~^ 

 and (make ji = 1) 



J 27t/>«, , 2t(j», , 27t«m 



X^. (i„ k^) y,. {k,, k^) 1 -7, (m„yfc,)r^ 1 Xv (m„/-,)«^= 2 y. (»/, -t) T*" • 



;?Jl=::=l rn2=l »i=l 



The first two factors occurring in these formulae, are according 

 to 4, III not equal to zero, because k^ and X-, are incommen- 



