Some examples of well-behaved Beurling number systems
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913702447415296 |
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| author | Broucke, Frederik Debruyne, Gregory Révész, Szilárd |
| author_facet | Broucke, Frederik Debruyne, Gregory Révész, Szilárd |
| contents | We investigate the existence of well-behaved Beurling number systems, which are systems of Beurling generalized primes and integers which admit a power saving in the error term of both their prime and integer-counting function. Concretely, we search for so-called $[α,β]$-systems, where $α$ and $β$ are connected to the optimal power saving in the prime and integer-counting functions. It is known that every $[α,β]$-system satisfies $\max\{α,β\}\ge1/2$. In this paper we show there are $[α,β]$-systems for each $α\in [0,1)$ and $β\in [1/2, 1)$. Assuming the Riemann hypothesis, we also construct certain families of $[α,β]$-systems with $β<1/2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_01567 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Some examples of well-behaved Beurling number systems Broucke, Frederik Debruyne, Gregory Révész, Szilárd Number Theory Primary 11N80, Secondary 11M26, 11M41 We investigate the existence of well-behaved Beurling number systems, which are systems of Beurling generalized primes and integers which admit a power saving in the error term of both their prime and integer-counting function. Concretely, we search for so-called $[α,β]$-systems, where $α$ and $β$ are connected to the optimal power saving in the prime and integer-counting functions. It is known that every $[α,β]$-system satisfies $\max\{α,β\}\ge1/2$. In this paper we show there are $[α,β]$-systems for each $α\in [0,1)$ and $β\in [1/2, 1)$. Assuming the Riemann hypothesis, we also construct certain families of $[α,β]$-systems with $β<1/2$. |
| title | Some examples of well-behaved Beurling number systems |
| topic | Number Theory Primary 11N80, Secondary 11M26, 11M41 |
| url | https://arxiv.org/abs/2309.01567 |