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Eigenvalues in the large sieve inequality, II

Olivier Ramaré (2010)

Journal de Théorie des Nombres de Bordeaux

We explore numerically the eigenvalues of the hermitian form ∑ q ≤ Q ∑ a mod * q ∑ n ≤ N ϕ n e ( n a / q ) 2 when N = ∑ q ≤ Q φ ( q ) . We improve on the existing upper bound, and produce a (conjectural) plot of the asymptotic distribution of its eigenvalues by exploiting fairly extensive computations. The main outcome is that this asymptotic density most probably exists but is not continuous with respect to the Lebesgue measure.

Exponential Sums with Farey Fractions

Igor E. Shparlinski (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

For positive integers m and N, we estimate the rational exponential sums with denominator m over the reductions modulo m of elements of the set ℱ(N) = {s/r : r,s ∈ ℤ, gcd(r,s) = 1, N ≥ r > s ≥ 1} of Farey fractions of order N (only fractions s/r with gcd(r,m) = 1 are considered).

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