On zero-density estimates and the PNT in short intervals for Beurling generalized numbers

TL;DR

This paper improves zero-density estimates for Beurling generalized number systems and explores their implications for the prime number theorem in short intervals, using advanced mean-value theorems for generalized Dirichlet polynomials.

Contribution

It provides significantly improved zero-density bounds for Beurling zeta functions and extends classical mean-value theorems to generalized Dirichlet polynomials, impacting prime distribution results.

Findings

Zero-density estimate: N(α, T) ≪ T^{c(1-α)/(1-θ)} log^9 T with c close to 4.

Enhanced understanding of prime distribution in short intervals for Beurling systems.

Extension of mean-value theorems to generalized Dirichlet polynomials.

Abstract

We study the distribution of zeros of zeta functions associated to Beurling generalized prime number systems whose integers are distributed as $N(x) = Ax + O(x^{\theta})$. We obtain in particular \[ N(\alpha, T) \ll T^{\frac{c(1-\alpha)}{1-\theta}}\log^{9} T, \] for a constant $c$ arbitrarily close to $4$, improving significantly the current state of the art. We also investigate the consequences of the obtained zero-density estimates on the PNT in short intervals. Our proofs crucially rely on an extension of the classical mean-value theorem for Dirichlet polynomials to generalized Dirichlet polynomials.