Quantitative results of the Romanov type representation functions Article Swipe

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YOU? · · 2022 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2204.12287

For $α>0$, let $$\mathscr{A}=\{ a_1(\log m)^α$ for infinitely many positive integers $m$ and $\ell_m<0.9\log\log m$ for sufficiently integers $m$. Suppose further that $(\ell_i,a_i)=1$ for all $i$. For any $n$, let $f_{\mathscr{A},\mathscr{L}}(n)$ be the number of the available representations listed below $$\ell_in=p+a_i \quad \left(1\le i\le \mathscr{A}(n)\right),$$ where $p$ is a prime number. It is proved that $$\limsup_{n\to \infty } \frac{f_{\mathscr{A},\mathscr{L}}(n)}{\log\log n}>0,$$ which covers an old result of Erd\H os in 1950 by taking $a_i=2^i$ and $\ell_i=1$. One key ingredient in the argument is a technical lemma established here which illustrates how to pick out the admissible parts of an arbitrarily given set of distinct linear functions. The proof then reduces to the verifications of a hypothesis involving well--distributed sets introduced by Maynard, which of course would be the other key ingredient in the argument.

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Combinatorics Mathematics Number theory Type (biology) Prime (order theory) Prime number Discrete mathematics Ecology Biology

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2204.12287
PDF: https://arxiv.org/pdf/2204.12287
OA Status: green
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DOI: https://doi.org/10.48550/arxiv.2204.12287

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Title: Quantitative results of the Romanov type representation functions

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Language: en

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Publication year: 2022

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Publication date: 2022-04-25

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Authors: Yong-Gao Chen, Yuchen Ding

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Landing page: https://arxiv.org/abs/2204.12287

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Concepts: Combinatorics, Mathematics, Number theory, Type (biology), Prime (order theory), Prime number, Discrete mathematics, Ecology, Biology

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