1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

Convexity Bound. For any c > 0, and 3tes = 1 /2, we have

(1.22) IL(n, s) I « Q7r(t)^1 /4+t:,

the implied constant depending on c and d.

As we shall see below, we refer to this bound as the convexity bound.

Remark 1.10. While we will often refer to this bound as the trivial bound for

L( n , s), it is, in this generality, not quite a trivial result: its proof uses implicitly the

heavy machinery of £-functions of pairs and a trick of Iwaniec to bypass the fact

that the Ramanujan/ Petersson Conjecture is not known in general. Note also that

there are several £-functions, like Rankin/ Selberg £-functions or triple product£-

functions, that are expected (but not proved so far) to be automorphic. In these

cases, the convexity bound is important for certain arithmetic applications and may

indeed be available, but at the cost of good bounds toward the RPC, such as the

deep results of Kim and Shahidi [KiSh, Ki] in direction of the Langlands functori-

ality conjecture; an exampple is the Rankin-Selberg £-function L(sym^2 n ® n', s) of

the symmetric square of a G L 2 ,q representation n times a G L 2 ,q representation n'.

Remark 1.11. From (1.26) below, one deduces easily (Rankin's trick) that

(1.23) L IA1r(n)I ~ xl+<: L 1;1~}1 «t: (xQ1r)t:x,

n~x n~l

for any x ~ 1 and any c > O; this is interpreted as the Ramanujan/ Petersson

Conjecture on average, and in many situations this bound is as good as RPC.

Proof. Our first step is the upper bound

(1.24)

for 3tes = 1 and any c > 0 with the implied constant depending only on c and d.

By the functional equation (1.1) and Stirling's formula

lr(s)I::::: lsl!Res-^1 /^2 exp(-~lsl),

2

one has, when ~es = 0,

(1.25)

Then for ~es= 1/2, (1.22) follows from (1.25), (1.24), and the Phragmen-Lindelof
principle.
To prove (1.24), we use the following argument of Iwaniec [13]. It was de-
signed to handle the case of the symmetric square £-function of a Maass form (for
which RPC is not available) and was extended by Molteni to more general auto-
morphic £-functions [Mol]. For simplicity, we present only a rough form of the
method, yet it is sufficient to handle the basic automorphic £-functions; we refer
to [Mol] for other related results. By convexity, it is sufficient to prove that for any
€,b > 0,

(1.26)