The eigenvalues of the quadratic form
V(φ, Q) = Σq∼Q
Σa mod q|S(φ, a/q)|2
are well understood when Q = o(N1/2), and this quantity is expected
to behave like a Riemann sum when N = o(Q). The behaviour in the range
Q∈ [N1/2, N] is still mysterious. In a previous work,
I have shown that the eigenvalues of this quadratic form have an L2-mean when
N/Q2 is not too large, leading to the conjecture that these eigenvalues
follow a distribution law.
Later studies tend to credit the idea that the density (if it exists!) is a highly non-trivial function.
This conjecture has been recently proved when N/Q2 is constant by
F. Bocca & M. Radziwiłł.
In the present work we investigate the other end of the range to be studied in Q and we show
in particular that we indeed have a limiting distribution when N/Q is as large as any power
of log N (and even a bit larger), and we present a full spectral analysis. We show in particular
that the quadratic form V(φ, Q) may not be asymptotic to
Q Σn |φn|2 when Q≈ N
but only on a vector space of positive but small dimension. We conclude this study by providing a ready-made
circle method depending only on the values of the trigonometric polynomials next to rationals of denominators
not more than some (small) parameter H and on values at points a/q where q is localized
around N/H and a is prime to q.