212 



Evidently the terms <?,; and 6, are supposed to be real: therefore 

 Mertens' theorem is only a special case of this theorem when the 

 terms of the series are real. It is however easy to see that Hahdy's 

 proof is also valid for tlie following extension to series with complex 

 terms : ') 



Theorem 4 ; //' S«„ converges absolutely to s, if bi -\- bi -\- . . . -\- bn 

 Is limited and Urn . b„ = 0, then the product o/ the series ^ai and 



?i = CO 



iY;, oscillates for n = cc about the same region as the series s. -Sft,,. 



The functions (p{n) and »K»,) are said to oscillate about the same 

 region if n tends to oo, if the following condition is satisfied: 



whatever be s^O we can find two numbers [i and a so that it 

 is possible to calculate whatever be n^ n a number m which 

 satisfies the conditions: 



\q:{n) — if'(")| <C * h* — m \ <Ca 



and that is also possible so calculate whatever be »i ^ (i a number 

 71 which satisfies the same condilions. 



Finally we prove the following theorem which is analogous to 

 theorem 4 and which contains theorem 1 as a special case: 



Theorem 5 : // Sa„ converges absolutely to s, if the mean-values 

 of order p of 2^b„ are limited and the mean-values of order (/) — 1) 

 [which we denote by UH''>) satisfy the condition: 



■ m) 



lim _:^=0, 



n = 00 7} 



then, the 7nea7i-7mlnes of 07-der p of the product-se/'ies oscillates about 

 the sa/ne regio7i as s. U''i' + ^^ as n te7ids to oo. 



Proof of theo7'e7n 4 

 Substituting /) = 1 in formula (7), we have: 



Hence, if 1 < ^' <C » = 



Wn rz: [a, «„ + ... + au ^,-fc+l] + {"k+\ iu~k + . • . + a„ «J = P + Q. 



Suppose \ti\ <Ct and \s,\ <ff whatever be i. 



1) It is not clear from Hardy's article how far the author also considers series 

 with complex terms; in the preceding pages he considers series with real terras, 

 and his statement, as far as I am aware, is also made for real terms; yet his 

 proof applies as well to series with complex terms. 



