Hal\'asz theorems for Gaussian ideals in sectors and short intervals

TL;DR

This paper extends Halász theorems to Gaussian ideals in sectors and short intervals, providing quantitative bounds and asymptotic formulas under non-pretentiousness conditions.

Contribution

It introduces a sectorial analogue of the Halász theorem for Gaussian ideals, including short-interval versions with new techniques like angular Fourier expansion.

Findings

Proves a quantitative Halász theorem for Gaussian ideals with bounds based on pretentious distance.

Establishes a sectorial asymptotic formula for sums over ideals in fixed sectors.

Develops a sectorial short-interval Halász theorem under non-pretentiousness and non-degeneracy conditions.

Abstract

We prove a quantitative Hal\'asz theorem for multiplicative functions on the nonzero ideals of $\mathbb{Z}[i]$, with bounds controlled by pretentious distance to the Archimedean characters $N^{it}$. We also prove a sectorial analogue: under angular non-pretentiousness, the sum of $f$ over ideals lying in a fixed sector is asymptotically given by the expected proportion of the unrestricted sum. Finally, under angular non-pretentiousness and a non-degeneracy condition on conjugate prime pairs, we prove a sectorial short-interval version of the Hal\'asz theorem for annular sectors whose radial thickness tends to infinity. The proof of the sectorial short-interval Hal\'asz theorem uses angular Fourier expansion, norm-compression to multiplicative functions on $\mathbb{N}$, and a theorem of Mangerel.