A new generalized prime random approximation procedure and some of its applications (Q6581841)

The authors of this paper prove the following assertion on Beurling prime systems.\N\NLet \(F\) be a non-decreasing right-continuous function such that \N\[\NF(1)=0,\ \ \lim_{x\rightarrow\infty}F(x)=\infty,\ \ F(x)\ll\frac{x}{\log x}.\N\]\NThen there exists a set of generalized primes \(\mathcal{P}=\{p_j\}_{j=0}^\infty\) such that \N\[\N\Big{|}\sum_{p_j\leqslant x}1-F(x)\Big{|}\leqslant 2 \N\]\Nand such that \N\[\N\bigg{|}\sum_{p_j\leqslant x}p_j^{-\mathbf{i}t}-\int_1^x u^{-\mathbf{i}t}\mathrm{d}F(u)\bigg{|}\ll \sqrt{x}+\sqrt{\frac{x\log(|t|+1)}{\log(x+1)}} \N\]\Nfor any real \(t\) and any \(x\geqslant 1\).\N\NIf, in addition, \(F\) is continuous, then the sequence \(\mathcal{P}\) can be chosen strictly increasing and such that \(\big{|}\sum_{p_j\leqslant x}1-F(x)\big{|}\leqslant 1 \).\N\NA new random approximation method is used to prove the main result of the paper. In addition, the authors present several applications of the proven main result to other problems related to the Beurling systems.