1549380232-Automorphic_Forms_and_Applications__Sarnak_

198 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS

Another property of the Landau/ Siegel zero, when it gets too close to 1, is

that it has the property to "repel" the other zeros (real and complex) from the line

~es = 1. This is illustrated in the following quantitative version (due to Linnik) of

the exceptional zero repulsion phenomenon discovered by Deuring and Heilbronn:

Exceptional Zero-Repulsion. There exists (effective) constants c 1 , c~ > 0 such that
for any T ~ 2 and any q ~ 1, if for some quadratic Xex( mod q), L(Xex, s) has an
exceptional zero
f3ex E [1 - ci/ log qT, 1],
then the product flx(q) L(x, s) has no other zero in the domain

'° , 1 _ c~[log((l -f3x. Jlog qT)l lc-. I / T·

:nes ::/

1

T , ::sms :::::: ,

ogq

here the product runs over all the characters of modulus q (including XexJ.

Finally, Siegel's famous theorem (in fact, a sharpening of a former result of

Landau) shows (ineffectively) that the exceptional zero cannot be too close to 1

Siegel. Let x be a primitive quadratic Dirichlet character of modulus q; for any c: > 0

there is a constant c( c:) > 0 such that L (x, s) has no zeros in the interval

[1 - c(c:)/qe, l].

Proof. Given 0 < c: < 1/ 16, one can assume that there exists some primitive qua-

dratic character, Xex (say) of modulus qex, having a real zero f3ex in the interval

]1 - c:, 1[ (otherwise we are done). This hypothetical zero is used to show that,

given any other primitive quadratic character x =f. Xex of modulus q, L(x, s) has no

zero within a distance c(c:)q-^4 e of 1. For this, one considers the auxilliary product

D(s) = ((s)L(Xex, s)L(x, s)L(XXex, s),

which has non-negative coefficients (since (1 + Xex(n))(l + x(n)) ~ 0). We proceed

as above and consider the integral

J = ~ J D(s + (3 )I'(s)xsds ="""' (1 * X * Xex * XXex)(n) e-n/x.

27fi x L.., nf3ex

(2) n~l

By positivity, the latter sum is bounded below by~ e-^1 /x » 1. We shift the contour

to ~es = 1/2 - f3ex < O; in the process, we pass a simple pole at s = 1 - f3ex with

residue r(l -f3ex)x^1 - f3•xL(Xex,l)L(XXex,l)L(x,l) and no pole at s = 0 (since

D(f3ex) = 1). The resulting integral is bounded by O(qqexx-^1!^2 +^1116 ); Hence, we

derive

1 « I= I'(l - f3ex)x^1 - f3ex L(Xex, l)L(xXex, l)L(x, 1) + 0( x 1 ; 2 ~~; 16 ).

We take x = q^3 and use the bound L(XXex, 1) « log(qqex) (which is easily derived

by a contour shift), to infer that

(logq)-lq-3e «xex,e L(x, 1),

where the implied constant depends on Xex and c:. The conclusion follows from

(1.19). We have followed the elegant presentation of Goldfeld [Gol] 0