On zero-density estimates for Beurling zeta functions
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916396091310080 |
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| author | Broucke, Frederik |
| author_facet | Broucke, Frederik |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10051 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On zero-density estimates for Beurling zeta functions Broucke, Frederik Number Theory Primary 11N80, Secondary 11M26, 11M41 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. |
| title | On zero-density estimates for Beurling zeta functions |
| topic | Number Theory Primary 11N80, Secondary 11M26, 11M41 |
| url | https://arxiv.org/abs/2409.10051 |