Gaussian almost primes in almost all narrow sectors (Q6620651)

The subset of \(\mathbb Z[i]\), the ring of Gaussian integers, lying in the first quadrant is denoted by \(\mathbb Z[i]^*\). The primes in there are precisely \(1+i\), the rational primes \(\equiv 3\pmod{4}\), and elements \(a+bi\) with \(a,b>0\) whose norm \(N(a+bi):=a^2+b^2\) is an odd prime. An important problem in the study of the distribution of Gaussian primes is to count how many there are in sectors. If the area is not too small one can give an asymptotic. If one considers only almost all sectors, the problem becomes easier. It is for example known that, given any \(\epsilon>0\), almost all sectors of area \((\log X)^{2+\epsilon}\) contain Gaussian primes under the assumption of the Generalized Riemann Hypothesis.\N\NDefine \(S_{\theta}(h)=\{n\in \mathbb Z[i]^*:N(n)\le X,\,\alpha\le \arg(n)\le \theta +h/X\}\). Put \(h_1=(\log X)(\log \log X)^{19.2}\) and \(h_2=(\log X)^{15.1}\). The authors show that almost all sectors \(S_{\theta}(h_1)\) and \(S_{\theta}(h_2)\) contain a product of exactly three, respectively two, Gaussian primes.\N\NThe proof strategy follows the approach of the second author to almost primes in almost all short intervals [Math. Proc. Camb. Philos. Soc. 161, No. 2, 247--281 (2016; Zbl 1371.11132)]. The problem is transformed into the problem of obtaining good bounds for mean square estimates of Hecke polynomials, which on its turns rests on a good understanding of Hecke \(L\)-functions. The required tools, such as an inequality of Halász-Montgomery type for Hecke polynomials, are developed first and then applied to obtain the main results.