Lecture 3. Large sieve inequalities
3.1. The large sieve
In this lecture we describe in greater detail the concept of quasi-orthogonality for
Hecke eigenvalues in families of automorphic forms: with the notations and as-
sumptions of Section 1.3.3, quasi-orthogonality means that
E(>.1f(m)>.1f(n), M·) = Om=n + ErrF(m, n)
where the error term ErrF(m, n) becomes small, as QF get large and m , n remain
in some range depending on QF. The orthogonality relations (2.20), (2.21) for
characters are an instance of quasi-orthogonality since, for 1 ~ m , n ~ q,
m=n(q)<==?m=n.The Kuznetsov-Petersson trace formula is another example for the range 1 ~ mn <
kq^4 , in view of (2.32). However, sharp bounds for individual ErrF(m, n) cannot
be obtained with sufficient uniformity in m , n to be truly useful for several impor-
tant applications (like the Subconvexity Problem), so one is inclined to mollify the
problem and to consider quasi-orthogonality on average. Thus we switch to the
problem of bounding, for N ;:;:: 1 as large as possible, the hermitian quadratic form
over CN
in terms of ll(an)ll = 'L-n~N lanl^2 • Such bounds that are valid for all (an)n~N
are called large sieve inequalities. For example, a version of quasi-orthogonality on
average would be a bound of the form
l I L an>.7r(n)l2dμF(7r) <<c: Q}:-L lanl
2
,
n~N n~N(3.1)valid for all a E cN and for N as large as possible; such a bound means that the
averaged contribution of the ErrF(m, n) is not much larger than the contribution
from the diagonal terms Om=n.
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