r 307 ) 



however large the mimher 7 may be, and taking this inequality into 

 account it is seen fVt>ni o(| nation (.1) liial we have necessarih- 



7', —JAni y'? =r 0. 



b,() b,o 



The proof given here is easily extended to tiie case in which // 

 is not ])rinie hnt a product of nnecpiai prime factors. Scpiare factors 

 are excluded, for then 7'/,^(, is identically zero. 



Let c be a prime number not dividing />, then instead of e(piatioii 

 (.1) we can establish the relation 



and it will appear that 7'/,,.,,) tends to zero, if we can show that 



Liia ^J'ij^ij=zO. Now the latter theorem is proved foi any [»iinie 



number A, hence it must remain true if repeatedly we mnltiplv // 

 by other prime factors c. 



In order to investigate the series 1\/, we consider the series of 

 functions 



^^ 1 — 2"'" 



Evidently it converges for \z\<^\. and expanding each term into 

 a power series, we find 



\l{ltl)Z 



Ltes.=i-i:''«" 



for the sum ^ ii{iï), in which the summation is extended over all 

 divisors d of y/^ unity and >u. itself incbuled, is zero except foi' //^ = J. 



Similarly we have by changing m into ml) 



'«=1 «4=1 (I I III 



Let h be the product of the prime factors //i,/^^ /^/., then 



:^(i{hd) is zero but for those integers /// that are of the peculiar form 



dint 



/' I\'- ' • /'/' ' •^"<' '" 'I'i^^ <*«^^^ ^ve have 2^" (, (M) = (i (h). 



djiii 



Hence we mav write 



