Acta Scientiarum Universitatis Pekiniensis (Naturalum)

HK 1958 46

34 lee oe

- The proof of the theorem is based on the following auxiliary lemma and two fundamental lemmata.

Auxiliary lemma (Page). There exists a constant C>0, such that in the region 1—Clog* D(|t|+1)<o <1, —w<i<+ce there are no zeros of any L(S,Z), (4%) with character % (mod D) excapt, possibly, one simple real zero of a

function Z (S,%) belonging to an exclusive real character (mod D). We can

~ ul prove that C> 180

First fundamental lemma. Let Z(S,%), Z(S, %)---L(S,%cp)-1) b3 all the £-functions balonging to a modulus D, let Q(D,wW) denote the number of Z-func-

for sufficiently large D.

tions each having at least one zero in the rectangle R 1—-log*D<o<l, |t|<e@”"Q5logD)+, (B) where W€[0,loglog D], U>0, then

5,020 e™, 0<w< a :

10% a”, es UD,W< ie

gC sonu) = Ui 2,

ef 223040) 2<wW<loglogD,

' Second fundamental Iemma. Suppos? that there exists an exclusive zero @ belonging to modulus D and let pp=6)+%t) b2 any zero of L(S,%), (X%X%), then

under the condition 1-6 <A, log D(| t)|+1), we have

Where 8=1—8, A;, Ay are suitable constants, and when D is large enough, we

1 lave A, D144 ° log*p »S log p 6 V pe. 4N? Ok : pRV*) p=!(modD)

the result required.

The result of the theorem may b2 improved, but it seems very difficult to reach the best result.

Here I should like to express my thanks to Professor Min Szu-hoa for his

help and encouragement.