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arXiv · 2211.08716

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

Frederik Broucke·Gregory Debruyne
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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.

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Frederik Broucke, Gregory Debruyne. 2022-11-16. On zero-density estimates and the PNT in short intervals for Beurling generalized numbers. https://doi.org/10.4064/aa221223-15-2

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