Mobius's Theorem on the Reversion of certain Series, 519 



Mobilises Theorem, § 3. 

 § 3. Let 



F ^ = (l-e a E a )(l-e b E b )(l-e c E c ). . /<*> ; 

 then, expanding the factors, we have 



F(aO = (l + e a E« + *«2E fl * + e a8 E a 3+ • • •) 



X(1+^E 5 + ^2E62 + ^ 3 E 6 3+ . . .) 



x(l + ^E c + e c2 E c2 + ? c3 E c3 + . . .) 



x /(*) 



=/(*) + te a f{x«) + 2^ a2 / (x a2 ) + $e ab f(x ab ) 



+ $e a3 f (V 3 ) + te a% f («*) + 2* a6c / («•*) 

 + . . . . 



where w has all values of the form a a b^c y .... The value of e\ 

 is supposed to be unity. 



Now from the given equation we have 



f(x)=(l~e a Ea)(l-e b E b )(l-e e E ) . . . F(*) 



= F(^)-2« a F(^) + 2«a6F(« oB )-2«« ft oF(0+ ■ • • 

 =S±«J(^*), 



where n has all values of the forms a, ab, abc, . . . and the 

 sign is positive or negative according as the number of factors 

 is even or uneven. 



We thus obtain the theorem: 



If a, b, c, . . . be any prime numbers, and if 



F(«)-S^/(«»), 



where the values of n are all the numbers of the form a a bPc v . ., 

 that is, all the numbers which have all their prime factors 

 included among a, b, c, . . ., then 



f( x )=t(-l)- en F( x % 



where the values of n are all the numbers of the forms 

 a, ab, abc, . . ., that is, all the numbers which have all their 

 prime factors included among a, b, c, ... and which are 

 divisible by no squared factor, and r denotes the number of 

 the prime factors of n. 



This is the most general form of the theorem published by 

 Mobius in Crelle's Journal, vol. ix. pp. 105-123*. The 



* " Ueber eine besondere Art von Umkehrung der Reihen." 



