( «t^ ) 



oil til 



page 305 of Vol YI. ''Series derived from the series 2 



Mathematics. — Rcnuxi-ks on the paper of Mr. Kluyver on 



II {my 



m 

 by Edmund Landau in Berlin. 



(Communicated in the meeting of May 28, 1904). 



In a paper recently published^) Mr. Kluyver treats the infinite series 



" (iimb + h) ^ti{h) ii{b + h) n{2b + h) 

 2^ mh-^h h ^ b+h "^ 2b-^h +•••'• • U 



7)1—0 



where b and h are two positive integers and where h can be regarded 

 as ^ h without limiting the generality. However, this research has 

 to be taken only in a heuristic respect ; it does not furnish the proof 

 that series (1) converges, i. e. that for every pair of values b, h 



li'T'T'-^ (2) 



1)1=0 



exists. *) 



The methods applied by me in the paper ') "On the prime num- 

 bers of an arithmetical progression" ("Ueber die Primzahlen einer 

 arithmetischen Progression") however allow such a proof to be made 

 which will be shown in § 2—7 of this paper, after I have reminded 

 my readers in § 1 of some theorems still known ; at the same time 

 we shall find an expression for the limit (2) i. e. the sum of the 

 infinite series (1) in finite form. In §§ 8—9 a conclusion is drawn 

 on the distribution of the numbers of an arithmetic progression 

 for which (i (/?) = + 1 resp. — 1 ; this justifies a supposition expressed 

 by Mr. Kluyver at the end of his paper. 



^1. Let /i {n), x, i'ii), ■■ ■ > Xr(i)(>^) be the (f{b) characters of the 

 group of the classes of residues modulo b prime to b, of which Xi(?^) 

 be the principal character (Hauptcharakter). If ?i and /; have a common 

 divisor we may understand by Xi(?^). '/aW. • • • > '/?iö) {n) the value zero. 

 Let moreover L,{s) for d = 1, 2, . . . , (f{b) denote the analytic 

 function determined by the Dirichlet series 



1) These Proc. VI p. 305. 



2) Only for the case = 1, h = l this was already known and from this ensues 

 directly, as Mr. Kluyver states at the beginning of his paper, the correctness of 

 the statement for any b and h = b, but not for the general case. 



3) Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, 

 mathematisch-naturwissenschaftliche Klasse, Bd. 112, Abt. 2% 1903, S. 493—535. 



