( 68) 



Mrs. Hadamard and de la Vallee Poussin have proved that no 



1 



Zv {s) possesses on the right line 3v (.y) =zl a zero, so that — — is 



regular for every v on the right line 9^(.y) = l. In the quoted paper 

 I have proved ^) the more general theorem : There is a positive number 

 « so that, when s = o -\- ti, in the region 



«>3, 1— <<y<2 



= log''t= = 



each of the <f {b) functions L, (.y) differs from zero and fulfills the 



inequality 



1 



§ 2. Now I denote by ]\fb,k (s) that analytic function which is 

 determined by the Dirichlet series 



00 



m—O 



convergent at least for a:=i!l{s)^l and I shall show that il/i,/i(ó') 

 can be brought in a very simple connexion with the functions Lj (.y). 



Let the greatest common factor {b, h) of b and h be put equal to 

 d. Without limiting the generality d can be regarded as being with- 

 out quadratic factor; for in the other case [i{mb-\-h) = 0, so every 

 member of the infinite series (1) is equal to zero. 



1. Let then be c/^1, so A prime iob. Then there is an integer Aj, 

 (determinate modulo b) for which 



/<j A = 1 {mod. h). 

 Now ensues from (3) for <? :^ 3v (.y) > 1 



_i__TTfi_2i!M^ = V^^^ (7) 



p «=i 



If we multiply (7) by Xv (^^) ^^^ sum up with respect to all 

 values of v we get 



V ^ = V X. (K) V ^^^^ = V '^ V x,(/vO. • (8) 



^^ L,{s) Jm^ ^^ n^ — ^ n^ ^-^ 



V=:l v=:l n=^\. W=l ■■'=1 



Now according to the fundamental property of the characters the 

 sum ^ Xv (0 differs from zero only, and then is equal to ^ (/v), when 



/ = ! (mod. h); hence 



1) I.e., page 521. Here I put the greater of the two numbers Cgi andc3i = «. 



