( 70 ) 



where / assumes all integer values ^ 0. In other words on the 

 right side of (10), when it is written short 



00 



li{d) ^ > lijk) 



d'^ ^^ k^ 

 L—i 



k assumes all positive numbers belonging to certain (> progressions 



of the form lb -\- h)., (P. = 1, 2, . . . , q), where <:^ It,. <^ b and 



{b, lij) = 1 ; so ' 



p 



ilA,;,(.)=^^J4/.. (..), (11) 



;=i 

 and from the result (9) of the case (I) follows after application to 

 the single members on the right side of (11) 



? 'ii^) 



1 nid) V ^ ^—1 1 

 Ml, ,, (.s) z= . LLJ \^ X . . . . (12) 



Of (1 2) the equation (9) is a special case, as for (/ = 1 

 B — h, II— h, 9 = 1, h, = />. 

 In I lie following the equation (12) may always be taken as basis. 



§ 3. The equation (12) proved above for <> ^ 1 furnishes in con- 

 nection with the properties of the functions L-j{.s) quoted in § 1 

 tirstly the analytic continuation of J/^^/t (.y) across the right line 

 (7=1. It teaches us that all points of the right line, .s = 1 included, are 

 regular places. From the theorem quoted at the end of § 1 follow^s more 



accurately that for t >• 3 and 1 — ^ ö ^ 2 the function 3fb /, (s) 



'' = log^ t — — 



is regular and satisfies there the inequality 



^^^ ''^^ <^ .1.9. (f{b) loff t = Q log^ t. 



i^'^wi^,;^*);?^ 



= 1 v=l 



U{s)\ ^<f{b) 



Let now a number tz >• « be chosen in such a way that in the 

 first place for any if ^ 3 we have 



and in the second place 7—— is smaller than the distance of the 

 ^ log^3 



right line (7 = 1 from any singular point of Mbh (•^')' t^^® imaginary 

 part of which lies between — 3/ and 3/. If moreover the equation 



\Mb,h {<J + ti)\ = \Mbj^ ((7 — ^01 

 is paid attention to, then ensues from the preceding : 



