228 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONSIn the case of Maass forms, one has a similar inequality (by comparison with
Weyl's law (4.20)) due to Deshouillers/ Iwaniec [DI]: fork= 0, 1 one has, for any
c:>O(3.8)The proof of (3.8) is similar to the proof of (3.7) above with Petersson's formula
replaced by the Petersson/Kuznetzov formula and with a more involved analysis of
the Bessel transforms, we refer to [DI] for a proof.Remark 3.3. It is interesting to take k = 1 and T = 0 in (3.8) above: then by
positivity and the identification (2.6), one obtains a large sieve inequality for holo-
morphic forms of weight 1:(3.9)We give below another more direct and more elementary proof (due to William
Duke) of this inequality which does not use the spectral decomposition of .6. 1.3.2. The Duality PrincipleA crucial ingredient for the proof of many large sieve type inequalities is the fol-
lowing duality principle:
Proposition 3.2. Let N ~ 1 and :F be afinite set, (A.,,(n)) rrEF any complex num-
l~n~N
bers and .6. > 0 a real number. Then the following are equivalent:
(1) For all (brr )rrEF E CF,L IL Arr(n)brr 1
2
~ .6. L I brr 1
2
.(2) For all (an)i~n~N E CN,LIL Arr(n)anl
2
~ .6. L lanl
2
.Proof. (1) states that the L 2 norm of the linear operator between Hilbert spaces
CF f--7 CN: (brr)rrEF f--7 (an)n~N := (L brr,\rr(n))n~N
rrEFis bounded by .6.. (2) states that the adjoint of the above operator is bounded by
.6.. The equivalence of (1) and (2) is then obvious. 0
The duality principle is quite powerful and flexible and can be applied in a
number of contexts where harmonic analysis is a priori missing. We illustrate this
with the derivation of two inequalities. The first is a direct derivation (due to W.
Duke) of the large sieve inequality (3.9) for holomorphic modular forms of weight
one. The second (due to Duke and Kowalski [DK]) deals with quite general families
of automorphic forms of arbitrary rank.