1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

The bound Hd(l/2) is probably far from the truth: assuming that the Functori-

ality Conjecture for Pairs holds, applying (1.6) to ?r^181 k gives the bounds Hd(l / 2k) for

7r at the unramified places (it holds for the ramified one too), and letting k ____, +oo,

one obtains

Ramanujan/Petersson Conjecture (RPC). Hd(O) is true.

Unconditionally, Proposition 1.1 can be strengthened to the following (non-

trivial) bound:

Theorem 1.1. For any d ~ 1, Hd(e) is true for

1 1

8=8d=----.

2 d^2 +1

We present below a proof due to Serre [Ser3]^1 that works for the non-archi-

medean places and uses the most basic analytic properties of L(7r ® 1T, s ); there

are also two alternative methods developped by Duke/ Iwaniec and Luo/ Rudnick/

Sarnak [Dul, Dul2, LRS, LRS2], which interrestingly both build on families of

twists L(x. 7r ® 1T, s) by appropriate characters. At the end of this lecture, we will

present the method of Luo/ Rudnick/Sarnak in the case of the archimedean place

(but the method works for every place and can be extended to number fields as

well).

Proof. The theorem is a consequence of the following refinment (due to Landau)
of Landau's Lemma [La] (see also [BR]):
Theorem 1.2. Let L (s) = L n .X(n)n- s be a Dirichlet L-series with non-negative
coefficients .X(n), and convergent for ~es sufficiently large. Assume that L(s) has
meromorphic continuation to C with at most poles of finite order at s = 0, 1; assume
also that L ( s ) is of bounded order in the half-plane ~es ~ -1, and that it satisfies a
functional equation of the form

q^8 L 00 (s)L(s) = wql-s L 00 (l - s) L (l - s)

for some constants w, q > 0, where

d

L 00 (s) = rrr(ais +{Ji),

i= l

for some d ~ 1 and ai ~ 0, f3i E C for i E {1 , ... , d}. Setting 'f/ = .Z::::f= 1 ai, one has
as x ____, +oo,

for all c > 0, where Pis some polynomial of degree ords=iL (s) and depends only on
L.
Applying this to L(7r ® 1T, s), one obtains

(1.10)

for some c7r > O; this yields

.A7r07r(N) = 07r,c(N^1 - J+^1 +c)

(^1) which might even go back to Rankin