Cg{n), and certain series of arithmetical functions 265 



The value ofcg(n). 



4. I show next that the value of Cg(») is given by (1*2). 

 Let us write 



O,00 = 2S/.(|) (4-1). 



If q and q' are prime to one another, we have 



C,(«)C,(«)=2M>(|)^(|). 



where 8 is a common divisor of q and n and 8' one of q' and n. 

 Clearly SS' runs through the common divisors of qq' and n, each 

 once only. Also q/S and q' /S' are prime to one another. Hence 



C,(n)Q(n)=|Z)/x(f'), 



where D = 88' runs through all common divisors of qq' and n ; 

 and so 



C,{n)CAn) = G,An) (4-2). 



Since Cg(7i) and Cg()i) each possess the multiplicative property, 

 we need only verify their identity when ^ is a power of a prime, 

 say CT*. Now 



p 

 w^here p runs through the -sr'-'"^ (ct — 1) numbers less than and 

 prime to -cr*^. These may be expressed in the form 



p = i^^~'^z + p', 

 where z = 0, 1,..., ct — 1 and p' runs through the numbers less 

 than and prime to is^~^. Hence 



and the sum with respect to z is zero unless -bj | n*. Thus 



c^,{)i) = {k>l,-sy{n) 

 and 6V (^i;) = t!r S e-2''2''-^/^*-^ = OTC t_iM. 



Now plainly 



c^ (n) = - 1 (zD- 1 ?i), c^ {)i) = in- - 1 (ot I w). . .(4-3). 

 Hence 



c^'' ('0 = (53- + w), c^2 (?0 = - CT (ot I ?i, tJT^ -f- n), 



Cto-^ (") = -OT (cj- - 1) (^2 1 n) ; 

 and generally 



c^* (n) = (ot^-1 + n), c^k (n) = - ct^-i (ot^-i | ?i, tn-^ + 7^), 

 c^,(n) = z7^-'(^-l) (w^|n) (4-4). 



* Following Landau, I write vr \ n for ' rar is a divisor of n ' and -ny + n for ' g; is 

 not a divisor of 7i,' 



