Sectorial Equidistribution of the Roots of x² + 1 Modulo Primes.

The article focuses on proving the joint equidistribution of normalized roots of the modular equation \(X^2 + 1 \equiv 0 \pmod{p}\) and the angles associated with primes \(p = a^2 + b^2\) of the form \(2\) or \(4n + 1\). Building on classical results by Duke, Friedlander, Iwaniec, and Hecke, the authors establish that the sequence of pairs—each consisting of a normalized root and its corresponding angle linked via non-inert prime ideals in \(\mathbb{Z}[i]\)—is equidistributed in the product space \([0,1] \times [0, \pi/2]\). Their approach employs an automorphic interpretation involving Poincaré series on arithmetic quotients of \(SL_2(\mathbb{R})\), addressing the novel challenge of nonspherical spectral analysis. Key technical achievements include constructing pointwise bounds for nonspherical Eisenstein series, utilizing a non-spherical analogue of the Selberg inversion formula, and applying spectral decomposition techniques to control associated Weyl sums. These results culminate in power-saving estimates for linear and bilinear sums, which, combined with a sieve method adapted from Duke, Friedlander, and Iwaniec, yield the main equidistribution theorem for primes.