Angles of Gaussian primes

Gaussian primes in almost all narrow sectors

“…Instead of asking about all short sectors, one can ask for the existence of prime angles in "typical" sectors, that is in almost all short sectors. Assuming GRH, Parzanchevski and Sarnak [9], and Rudnick and Waxman [11] showed that almost all sectors contain a prime angle, in fact that the asymptotic formula (1.1) (at least in a smooth form) remains valid for almost all I if |I| > X −1+ε , which is the most we can expect (up to log factors) since the number of prime ideals with norm about X is roughly X/ log X, hence almost all sectors shorter than 1/X will contain no prime angles θ p with N(p) ≈ X. Rudnick and Waxman [11] gave a precise conjecture about the asymptotic behavior of the number variance, supported by a theorem for a function field analogue of the problem.…”

Sup-Norm and Nodal Domains of Dihedral Maass Forms

“…To prove this claim, we need an interesting elementary result on the repulsion of angles for a real quadratic field (Lemma 15), which shows that angles are well-spaced. This repulsion between angles is also an important fact when one studies angle distribution in short arcs (see [35] for the imaginary quadratic field Q(i)).…”