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

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 \[ N(α, T) \ll T^{\frac{c(1-α)}{1-θ}}\log^{9} T, \] 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.

22 pages, fixed some minor errors

References in corpus (2)

• A new generalized prime random approximation procedure and some of its applications
• The Carlson-type zero-density theorem for the Beurling zeta function

Cited by in corpus (1)

• A Mean Value Theorem for general Dirichlet Series