1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. LARGE SIEVE INEQUALITIES

Proposition 3.3. there exists constants b, c > 0 such that for 3/4::::; o: < 1

L N(sym2 Ex; o:, T) « yb xc(l-u)

xEZ,6.(x ) # 0

1x 1 :;;;x

here b, c and the implied constant depend only on A , B.

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One can use this to study the distribution of the special values {L(sym^2 E x,1),
lxl ::::; X}; in this case the quasiorthogonality formulas come from the Lefschetz
trace formula and Deligne's equidistribution theorems [De3], applied to the reduc-
tion of Et modulo almost all primes and to its symmetric powers. Hence one can
prove:
Proposition 3.4. Assume also that A(t), B(t) are coprime; as X ---> +oo on the space
[- X, X] n Z, the random variables x ---> L(sym^2 E x, 1) and x ---> L(sym^2 Ex, 1)-^1
admit a limiting distribution function.
Remark 3.9. The above example illustrates the fact that neither a "dense" family4,
nor a strong form of quasi-orthogonality is necessary to establish the existence of a
distribution law for the values of L-functions near 1.
There are many other zero density estimates of different sorts, depending on
the kind of L-functions or on the shape of the region in which zeros are considered.
Also, many refinements are possible for more specific families for which stronger
large sieve inequalities are available. We have not touched here on more advanced
sieving techniques that would allow such results -like the mollification technique.
For other kinds of zeros density estimates, the reader may look at [Mon, Bo, HJ]
or [KM2, HB-M]. To date, the most striking zero density estimate is probably the
following bound of Conrey/Soundararajan [Cons]:
Theorem. The following bound holds:
L N(xo; 0, 0) ::::; (c + o(l)) L 1, X---> +oo,
10 1::;;x 101 ,,;x
where D ranges over fundamental discriminants of quadratic fields, XD denotes the
associated quadratic character, N(x o ; 0, 0) the number of non-trivial (i.e. within the
critical strip) real zeros of L(xo, s), and c is a constant strictly smaller than 1. In
particular (since c < 1), there is a positive proportion of fundamental discriminants
D such that L(xo, s) has no non-trivial real zeros!

(^4) In terms of the size of the conductors compared with the size of the family: for lxl ::::;: X , the conductor
of L(sym^2 E x, s ) is typically ~ X^2 deg.::> where /::,. is the discriminant, while the nu.mber of such x is
« X.