246 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONSthe trivial bound being (at least when the sequence (an) is not lacunar).C(no,N) «,, Q~ 0 (~.= lanl) «c: Q~ 0 N^112 (L lanl
2
)^112.To achieve this one may wish to apply the method of moments and average
over an appropriate family F containing n 0. Taking a moment of order 2k, the
discussion at the end of the first lecture shows that, by multiplicativity of the ,\7r ( n),
this amounts essentially to bounding the mean square of some linear form of length
Nk:l~I L I L bn,\7r(n)l2·
7rEF n~Nk
If Nk is not too large, one can expect that the harmonic analysis on Fis sufficiently
rich to imply quasiorthogonality in this range, namely that(4.4)We then get by positivity.C(no, N) «,, (QFN)"IFl1/2k( L lanl2)1/2.
n~N
Hence, if we can take k large enough (namely k > k 0 := log IFI/ log N), we can
improve over the trivial bound. Observe that all the large sieve type inequalities
encountered so far give the bound ( 4.4) for k less than ko and no larger, thus giving
no non-trivial bound for the original individual sum. This shouldn't be a surprise,
since the large sieve inequalities are very general and make no use of the particular
arithmetic properties of the sequence (an); we have experienced the strength of
this generality in the last lecture and we discover its weakness now. In fact, there
is no chance to go beyond the trivial bound without using some peculiar features
of an: perhaps the most convincing example is to consider an = 3:71" 0 ( n)!
For the Subconvexity Problem, the fundamental property of (an) is smoothness,
since an is essentially of the form
1 n
an= r,;; V( ~),
yn V Q7rO
for some smooth rapidly decaying function V. With such an extra assumption, one
may hope to get a bound similar to ( 4.4) for k > ko. Indeed, there are a priori two
possibilities available for achieving this:
(1) either one keeps F unchanged and one increases k beyond the critical
moment k 0 and one tries to obtain (4.4);
(2) one sticks to k = k 0 and one shortens F to decrease IFI.
The first possibility is not very flexible since k has to be an integer so must be
increased by at least one: then the problem of obtaining ( 4.4) for the resulting
form of length Nko+l , even for smooth an, may well be beyond the capacities of
the current technology. (However, note that if for all n E F the linear form L(n, N)
is non-negative, there is still the possibility of increasing k by only half an integer
and this may be short enough to yield sharp bounds [Conl] .)