Y. 



( 67 ) 



n" 

 «=1 



As is known this series converges, in case v = l, for 3t (.v) ^ 1, 

 and in case v = 2, . . . , (f{h), for i)v is) > 0. 



The equation holding good for 3i {s) > 1 and every v {= 1, . . . , if.{b) ) 



L,(.,) = V^ = n— TT (3) 



^^ n' -L -L Xv(p) 



f pS 



where p passes through all prime numbers, shows tiiat no L, {s) 

 possesses a zero with real part ^1. The equation (3) gives for r = 1 



A«=ri-TiiO-,T)=^(')riO-,l) • (^) 



ƒ 



where p passes through all prime factors of ^. From (4) it follows that 

 L^ [s) may be continued across the right line i)i [s) = 1 and that it 

 possesses in s = 1 a pole, so that 



1 



li7n :=: (5) 



Further Dirichlet ^) has expressed the quantities 

 ^2 (1) = > , • . • , %ö) (1) = > 



n .^m^ n 



«=1 »i=:l 



in finite form by logarithms and trigonometric functions and proved 



moreo^'er — what did not at all ensue from it — that each of 



the afore mentioned (p(J)) — 1 quantities is different from zero. So 



the limits 



,,11 .1 1 



lim =: 1 ' • • 1 Ivm = . . . (6) 



s=i ^2(«) ^2(1) s=\ L^[b) (s) L.^{b) (1) 



do exist. 



^) "Proof of the theorem that every unlimited arithmetical progression of which 

 the first term and difference are integers without common factor, contains an 

 infinite number of prime numbers," Transactions of the Royal Prussian Academy of 

 Sciences at Berlin, 1837, p. 45—71; Works, Vol. 1, 1889, p. 313—34^2. 



(„Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren 

 erstes Glied und Difïerenz gauze Zalilen olme gemeinschafllichen Factor sind, 

 unendlich viele Piimzahlen enthalt", Ablumdlungen der Königlich Preussischen 

 Akademie der Wissenschaften zu Berlin, 1837, S. 45—71; Werke, Bd. 1, 1889, 

 S. 313—342). 



5* 



