Mobilises Theorem on the Reversion of certain Series. 529 



we see at once that the expression in question is equal to 

 %«/(^), where 



-44X'-3(-D-. 



n having the same values as before, and a,b,c,... being the 

 prime factors of n. 

 § 16. If e n =n, then 







=-KX>-DK>- 



This is the well-known expression for the number of numbers 

 less than n and prime to it. Adopting Gauss's notation and 

 denoting this quantity by <f>(n), we thus obtain the result : — 



(1-E t )(i-E,)(l-E,)... „. 



(1-2E 2 )(1-3E 3 )(1-5E 6 )../ W 

 =/(.*) + <j> (2)/(^) + £(3y«) + <f>(i)M) + &o- 

 § 17. As a particular case of this theorem, let 



we thus find 



(1-E,)(l-E,)(l-E t )„. g 

 (1-2E 2 )(1-3E 3 )(1-5E 5 )... 1-a 



__ * »(2>> »(3)^ 0(4)^ 

 Now 



1 _x_ 



(1-2E 2 )(1-3E 3 )(1-5E 5 )... 1-x 



x 2x 2 Sx 3 4^ 4 



= i + i 2+1 i+n 4+ &C. 



1— X 1— 2 1— /B 3 1— X* 



=^ + cr(2> 2 + <7(3)^ 3 + <7(4> 4 + &c; 

 and, operating on this series with 



(1-E 2 )(1-E 3 )(1-E 5 )..., 

 we obtain as result 



tl<r{n)x n -€ l o-(n)^ n -Si <r(n)x 3n 



S 1 cr(n)x bn + % 1 <r(n)x — &c. 

 Also it is evident that 



(l-E 2 )(l-E 3 )(l-E.,)... T ±-= ;C ; 



