279 



a I 



Pr, 





or 



« / 



ƒ (,) ±= 77 I 1 + :^ 



m=I \ 1=1 ip' )' 



or, what comes to the same thing, tor any Dirichlet's series for 

 wliicli tlie connected power-series in an i. n. of v. has one of tiie 

 forms 



P{w^,a!„...:c,„,. ..) = Jq„(*„) ..... (2) 



or 



P (.V,, x„... .v,„ ,...)= no +Q„ (.r,,)) ... (3) 



where (j„ (a.'„) (/i = 1, 2, . . . .) is a power-series in x,, withont a con- 

 slant term. The eqnalitj B ^ D is a consequence of tiie theoren) : 

 If: a. The series is bounded^) for |a,;J ^ On {n ^1,2,... .), tlien 

 f). it is absolutely convergent for \x„\<^^G,i, where d is an 

 arbitrary positive number in the interval <-' (9 <^ 1 '). 



Now, if we consider the power series (2) and (3), we see that 

 the variables x„ occur to some extent separated from one another. 

 This led Bohr to the conjecture, thai the equality B^D would 

 hold for any Dirichlet's series, for whicii I lie variables in the con- 

 nected power-series in an i. n. of v. do not occur too much mixed up. 

 Con tir mat ion of this conjecture is the purpose of the present com- 



') According to Hilbert (Wesen unci Ziele elner Analysis der unendlich vielen 

 unabhiingigen Variabeln, Palermo Rendiconti, vol. 27, p. 67) a power-series in an 

 i. n. of V. is defined to be bounded if: 



1". The power-series Pm (Xi, .Cj, ... .):„,) (AbschniUe), that may be obtained from 

 the power-series in an i. n. of v. by putting .Cni-j-i — .x'in-|-2 = . . . = 0, are, for all 

 values of m, absolutely convergent in the region |.i'il^ Gj, \Xi\ $ Gji • • • • l^ml ^ Grm- 



2". There exists a number K, independent of m, such that, for every m, the 

 inequality 



holds in the region |;ri| ^ ffj, \Xc,\ ^ Go, . . . . \xm\ ^ Gm- 



^) It is well known, that b follows from a for any power-series in a finite 

 number of variables. Originally Hilbert had assumed this also, as being self evident, 

 for an i. n. of v. But Bohr showed that this could not be true by constructing an 

 example to the contrary. 



