Mobius's Theorem on the Reversion of certain Series. 523 



where n denotes any number, and a, 6, c, . . . are its prime 

 factors. 



§ 7. These formulas hold good for all values of r, including 

 r=0 and r negative. When r=l, <r r (n) becomes cr(n), the 

 sum of the divisors of n, and the formulae become 



*M-2^G)+^(|)-S«&«r(j- C ) +• • - = 1, 



When r=0, cr r (V) becomes v(n), the number of the divisors 

 of n } and the two formulae coalesce, each reducing to 



K»)-^(0 + ^5)- S v(£) + ...=i. 



Formula involving A r (n), § 8. 



§ 8. Let e 2 = and e p =p r , p being any uneven prime, and, 

 as in the last section, let 



Then 



w v * 3'a 8 , 5V 7V , . 



= ^ + A,(2> 2 + A r (3> 3 + A r (4> 4 + &c, 



where A r (w) denotes the sum of the rth powers of the uneven 

 divisors of n. 

 Thus we have 



f(x) = x+ x 2 + x d + x^ + &c, 



</>(#)=# + 3 r # 3 + 5 V + 7 r ^ 7 + &c, 



F(«) = x + A r (2> 2 + A r (3> 3 + A r (4> 4 + &c, 



e 2n =0, e 2n+1 = (2n + l) r , y n =l; 



and by substituting these values in the formulae of § 5, and 

 equating coefficients as in the last section, we obtain the 

 formulae : — 



A r ( re ) -SA'A^£) + SA'B'A^) - . . . = 1, 



= or n r ', according as n is even or uneven ; 



where A, B, C, . . . are used to denote the uneven prime factors 



2M2 



