280 



raiinication. In fact it can be proved that B^D holds for any 

 Dirichlet's series tliat can formally be written in tiie form 



-.=1 i=\{y„y 



wliere rp is an arbitrary (non-constant) ') integral function. As a 

 consequence of the relation, already mentioned above several times, 

 the following theorem concerning power-series in an i. n. of v. is 

 equivalent to this statement. 



Theorem. If (f is an integral function and Q„(.r,) [nz=\,2, ...) a 

 formal ') power-series in .r„, without a constant term, and if the 



power-series in an i. n.ofv. P(j;,,,r, x,,, )=t(Qi('''i) + QiC-^'i) 



+ + Qm(''m) + •••■) 's bounded for |x„j ^ ö„ (n = 1, 2, . . . .), 



then it is absolutely convergent for \.r„\ ^6 G,,, ifO<^(9<Cl. 



In the following pages an outline of llie proof of this theorem 

 will be given. 



^2. For the sake of simplicity we take 6^, = G, =....=r6r„ = 

 = 6'>1, but dG<^\. 



Because the given power-series in an i. n. of v. is bounded, there 

 exists a number A', not depending on 7/i, such that 



I V' (Q, l-^,) + Q, (*,) + • ■ + C>m (■?« )) < ^. . . • (-1) 

 The first part of the proof of the theorem of ^ 1 discusses the 



power-series Q«(^»)(»=1.2 )• It is proved that it follows from 



(4) that all these power-series possess a certain region of conver- 

 gence. Further research shows that two cases may occur: 



r. The functions Qn{x„) are all regular for |.r„|< ö. This is the 

 general case. 



T. If the integral function ff{y) has the form F\«''^/ (where V 

 is again an integral function), then it is only possible to conclude 

 that the functions QniXn) are logarithms of functions regular for 

 \xn\<^G, namely that they have the form Q„{x„) = log{l -f- /?„(•!•„)), 

 where Rn{x„) is regular for |.r„|<^ 6^, and R„{0) = 0'). 



1) If p is a constant, the theorem is trivial. 



') That is to say, the existence of a region of convergence is not assumed, but 

 will appear to be a consequence of the other assumptions. 



') It is interesting to observe, that obviously the series (2), with ip{y) = y, 

 falls under the first case, and the series (3), with f{y) = e^, Viz) = z, under the 

 second case. 



