1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

variants are called the Subconvexity Problem in the t-aspect, the level-aspect or the

oo-aspect, respectively.

As we shall see in Lecture 5, the solution of the ScP in any of these aspects

has striking consequences, but let us start first with a very basic and very practical

corollary of (sub)convexity: consider a fixed, smooth, compactly supported func-

tion V on R>o. For X? 1 we consider the problem of bounding the sum of length

'.:::'.X,

~v(7r, X) = L ·Aoor(n)V(;);

n;;;, 1

our goal is to improve on the "trivial" bound

~v(7r, X) «,,,v (Q1l'X)" X ,

which follows from (1.23). By the inverse Mellin transform, we have

~v(7r, X) = -.^1 j L(7r, s)V(s)X,^8 ds, where V(s) , = 1= V(x)x^8 dx -.

27ri 0 x

(3)

Now one may shift the contour to ~es = 1 /2, passing no pole in the process (unless

L(7r, s) = ((s)), and we obtain eventually (integrating by parts several times in

V(s) to gain convergence) that for any A > 0,

(1.29) ~v(7r, X) «v,A x^1 /^2 sup

1Res=l/2

« xl/2 sup Q7l' l/4- o+ c (t) « Q" xl/2Ql/4-8

V ,A ,c I IA V,A,c 7l' 7l' '

1Res=l/2 S

for some J? 0. When J = 0 (i.e. for the convexity bound), we already see

an improvement over the trivial bound ( Q1l'X)" X for X in the range X » Q;/^2 +",

and a subconvex exponent would provide an improvement for X in the larger range

X » Q; 12 -^28 ; ultimately, GLH would provide an improvement for all X »e Q~

for any c > 0.

Remark 1.12. This kind of improvement over (1.23) is the manifestation of the

oscillation of the coefficients >.11' ( n) in the given ranges of X: indeed, it follows

from GRH for L(7r ® ii', s) that for any X? 1 and any c > 0,

L i>.7l'(n)llV(;)I »,,,v (Q1l'x)-"X.

n~x

1.3.2. Interlude: How to compute L( 7r , s) inside the critical strip?

As we are interested in the behavior of L(7r, s) within the critical strip 0 <~es< 1,
we should first have a manageable expression for L(7r, s) to work with; this is
not immediate since s = a + it is not within the region of absolute convergence.
However, one can obtain such an expression by standard means using the functional
equation and contour shifts: the result is called (sometimes inappropriately) an
"approximate functional equation". There are various possibilities for achieving
this, the choice depending on the problem considered; we follow here a derivation
borrowed from [DFI8] (but see also the elegant version of [Hal]). One starts with