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arXiv · 2409.10051

On zero-density estimates for Beurling zeta functions

Frederik Broucke
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Abstract

We show the zero-density estimate \[ N(ζ_{\mathcal{P}}; α, T) \ll T^{\frac{4(1-α)}{3-2α-θ}}(\log T)^{9} \] for Beurling zeta functions $ζ_{\mathcal{P}}$ attached to Beurling generalized number systems with integers distributed as $N_{\mathcal{P}}(x) = Ax + O(x^θ)$. We also show a similar zero-density estimate for a broader class of general Dirichlet series, consider improvements conditional on finer pointwise or $L^{2k}$-bounds of $ζ_{\mathcal{P}}$, and discuss some optimality questions.

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Frederik Broucke. 2024-09-16. On zero-density estimates for Beurling zeta functions. https://arxiv.org/abs/2409.10051

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