Mathematics. — "A theorem concerning power-series in an infinite 

 number of variables, with an application to Dirichi.et's ') 

 series." By H. Ü. Kloosterman. (Communicated by Prof. 

 J. C. Kluyver.) 



(Communicated at the meeting of March 24, 1923). 



^ J. An important relation between the theory of Dirichi.et's 

 series and the llieoiy of power-series in an infinite number of variables 

 (for abbreviation we shall write: power-series in an i. n. of v.) has 

 been discovered by H. Bohr'). Let 



/(.) = i "^ , s = a + it (1) 



be an ordinary Dirichlet's series. Put^Ei = — , .r, =~ , x,„ = 



1 



= — , . . . (where p,„ is the 7?i-th prime-number, and let n ^= p"' //a . . . /jV, 



where /)„, , z»,,, , . . . />„,. are the ditferent primes which divide «. 

 Then the series (1) can formally be written as a power-series in an 

 i. n. of v., thus : 



00 



P (.«,, .«,, . . . Xm ....)= i' a„ «"I X'i . . . .«'r = 



n=l ' 



C -)- ^ Ca ,V.j, + ■2' Ca„? Xa Xji + JE Ca,^,y «„ X;i X-j -\- . . . 

 u.=ï,'l,... a,3=l,2,.. a,,3,v=l,2,... 



a < j8 "■<?<'' 



This relation has been applied by Bohr to the so-called absoiute-con- 

 vergence-probieni for Dirichlet's series, tliat is to say the determ- 

 ination of liie abscissa of absolute convergence of (1) (the lower 

 bound of all numbers {i, such that the series (J) converges for 

 'J > ?, '" terms of (preferably as simple as possible) analytic 

 properties of the function represented by (1). Let B be the abscissa 

 of absolute convergence of (1), and Ü the lower limit of all numbers 

 a, such that f{s) is regular and bounded for o^a. The absolute- 

 convergence-problem will be solved, if the difference B — D is 

 known. Bohr proves that B=D for any Dirichlet's series that 

 can be formally represented in one of the following forms: 



') A more detailed proof of the theorem will be published elsewhere. 

 2) Göttinger Nachrichten, 1913. 



