POSITIVE PROPORTION OF SHORT INTERVALS CONTAINING A PRESCRIBED NUMBER OF PRIMES
POSITIVE PROPORTION OF SHORT INTERVALS CONTAINING A PRESCRIBED NUMBER OF PRIMES
复制标题
包含规定数量素数的短区间的正比例
DOI：
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发表时间：
2018
影响因子：
0.7
通讯作者：
Daniele Mastrostefano
中科院分区：
文献类型：
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作者：
Daniele Mastrostefano
We prove that for every $mgeq 0$ there exists an $unicode[STIX]{x1D700}=unicode[STIX]{x1D700}(m)>0$ such that if $0<unicode[STIX]{x1D706}<unicode[STIX]{x1D700}$ and $x$ is sufficiently large in terms of $m$ and $unicode[STIX]{x1D706}$ , then $$egin{eqnarray}|{nleq x:|[n,n+unicode[STIX]{x1D706}log n]cap mathbb{P}|=m}|gg _{m,unicode[STIX]{x1D706}}x.end{eqnarray}$$ The value of $unicode[STIX]{x1D700}(m)$ and the dependence of the implicit constant on $unicode[STIX]{x1D706}$ and $m$ may be made explicit. This is an improvement of the author’s previous result. Moreover, we will show that a careful investigation of the proof, apart from some slight changes, can lead to analogous estimates when allowing the parameters $m$ and $unicode[STIX]{x1D706}$ to vary as functions of $x$ or replacing the set $mathbb{P}$ of all primes by primes belonging to certain specific subsets.
