1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

or the bounds for I given in (2.30), one can derive the following approximated
versions of the above identities. For holomorphic forms of weight k ~ 2, one has
(here the constant implied is absolute):

r(k-1) ~ -

(2 .32) ( 4 7r)k-l ~ .Jmnp 1 (m)pt(n) =

fEBk(q,x)

1/ 2 2 (mn)l/ 4

Om=n + O(T(q(m, n))(m, n, q) log (1 + mn) ..Jk );

q k

for Maass forms of weight k = 0 or 1, one has:

(2.33) .Jmn2:H(t1tP1(m)p1(n) + L J_ r H(t)75a(m, t)pa(n, t)dt =

j~l a 47r JR

(mn)l/ 4

CAOm=n + OA(T(q(m, n))(m, n , q)^112 ).

q

Hence with an appropriate averaging, the Fourier coefficients { /npt(n)} fEB(q,x)
are approximately orthogonal in the sense of section 1.3.3.

Remark 2.3. Applying (2.32) or (2.33) to m = n and using positivity, we conclude
that for f a cusp form, one has

(2.34) VnPJ(n) « c;,J nl/4+£

for any c > 0. We will refer to this bound as a Kloosterman type bound; by applying
it to a primitive form f and using the multiplicative properties of Hecke eigenval-
ues, we conclude that
l>-1(n)I:::;: T(n)n^114 ,
which is H 2 (1/4) for the finite places. There is a similar argument for the infinite
place (with a somewhat different test function H(t)), which proves H 2 (1/4) for the
infinite place (this is Selberg's bound). Note also that any improvement over the
trivial bound for Kloosterman sums yields H 2 (B) for some e < 1/2.