论文标题
罗曼诺夫类型表示功能的定量结果
Quantitative results of the Romanov type representation functions
论文作者
论文摘要
对于$α> 0 $，令$$ \ MathScr {a} = \ {a_1 <a_2 <a_3 <a_3 <\ cdots \} $ {$ \ mathscr {l} = \ {\ ell_1，\ ell_1，\ ell_2，\ ell_2，\ ell_2，\ ell_3，\ ell_3，\ cdots \ cdots \ cdots \ cdots \ cdots \ cdots \ cdots \ cdots \ cdots。带有$ \ mathscr {a}（m）>（\ log m）^α$的正整数，用于无限的许多正整数$ m $和$ \ ell_m <0.9 \ log \ log \ log m $，用于足够整数$ m $。进一步假设所有$ i $ $（\ ell_i，a_i）= 1 $。对于任何$ n $，令$ f _ {\ Mathscr {a}，\ Mathscr {l}}}（n）$是$ \ ell_in = p+el_i \ el_i \ quad \ quad \ left（1 \ le le i i i \ le i \ le \ le \ mathscr {a}（a a}（a a}（a a}（a a}（a a}（a a），事实证明，$ \ limsup_ {n \ to \ infty} \ frac {f _ {\ mathscr {a}，\ m缩{l}}（l}}（n）} {\ log log \ log \ log \ log n}> 0，$ $覆盖了Erd \ h os__i $ n $ n $ n $ n $ n $ a y_i的旧结果，该论点中的一种关键要素是这里建立的技术引理，它说明了如何挑选一组任意给定的不同线性函数的可接受部分。然后，证明降低了涉及梅纳德（Maynard）引入的良好分布集的假设的验证，这当然是该论点中的另一个关键要素。
For $α>0$, let $$\mathscr{A}=\{ a_1<a_2<a_3<\cdots\}$$ and $$\mathscr{L}=\{ \ell_1, \ell_2, \ell_3,\cdots\} \quad \text{(not~necessarily~different)}$$ be two sequences of positive integers with $\mathscr{A}(m)>(\log m)^α$ for infinitely many positive integers $m$ and $\ell_m<0.9\log\log m$ for sufficiently integers $m$. Suppose further that $(\ell_i,a_i)=1$ for all $i$. For any $n$, let $f_{\mathscr{A},\mathscr{L}}(n)$ be the number of the available representations listed below $$\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.
