On zero-density estimates and the PNT in short intervals for Beurling generalized numbers
Volume 207 / 2023
Acta Arithmetica 207 (2023), 365-391
MSC: Primary 11M26; Secondary 11M41, 11N05, 11N80.
DOI: 10.4064/aa221223-15-2
Published online: 11 April 2023
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 $$N(\alpha , T) \ll T^{\frac{c(1{\textstyle-}\alpha )}{1{\textstyle-}\theta }}\log^{9} T$$ for a constant $c$ arbitrarily close to $4$, improving significantly the current state of the art. We also investigate the consequences that the zero-density estimates obtained have 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.
