Some problems of analytic number theory on arithmetic semigroups
Let \(E\) be a set of primes satisfying \[ (*)\quad \sum_{ p \in E, p \leq x} \frac{\log p}{p}=\tau \log x +O(1), \] for some positive constant \(\tau\). Let \(A\) be the multiplicative semigroup generated by the primes from \(E\). The authors study a variety of natural questions on the distribution of elements in \(A\), or multiplicative properties of elements of shifted sets \(A+k\). Questions on the counting function of \(A\) go back, for example, to \textit{E. Wirsing} [Arch. Math. 7, 263--272 (1956; Zbl 0075.25001)]. Questions on prime factors of shifted members of \(A\) have been raised for example in [the reviewer, Funct. Approximatio, Comment. Math. 35, 61--77 (2006; Zbl 1196.11139) and Bull. Lond. Math. Soc. 40, No. 1, 97--107 (2008; Zbl 1203.11064)]. Here we state some of the new results. A results for short intervals: Let \(1 \geq \tau > \frac{1}{2}\) and suppose that \((*)\) holds: then there exists \(\delta=\delta(\tau)>0\) and \(C(E)>0\) such that \[ \left|\{n\in A, x \leq n < x^{1-\delta}\}\right|>\frac{C(E)x^{1-\delta}}{(\log x)^{1-\tau}}. \] A result for shifted primes: Let \(k\neq 0\) be an even integer. There exist \(\tau_0<1\) such that if \(\tau_0 \leq \tau \leq 1\) and \(E\) satisfies \((*)\), then there exists \(C(E)>0\) such that \(|\{p+k\in A, p\leq x\}|> \frac{C(E)x}{(\log x)^{2-\tau}}\). If \(2 \in E\), this also holds for odd \(k\). A result on large prime factors of shifted elements: Suppose there is some constant \(B>0\) such that \(\sum_{ p \in E, y \leq p <2y} 1\gg \frac{y}{(\log y)^B}\). Let \(k\neq 0\) be an integer and \(\varepsilon >0\). Then there exists \(C(E)>0\) such that there are at least \(C(E) \frac{x}{(\log x)^{3B}}\) solutions to \(m \in A\), \(m+k=rp\), \(p>x^{3/5 - \varepsilon}\). Moreover, the size of the prime factors of \(m\) can also be restricted: Let \(P(n)\) denote the greatest prime factor of \(n\). Then there are infinitely many solutions to \[ m+k=n,\quad P(m)< \exp\left( (\log m)^{\frac{1}{2}+\varepsilon}\right),\quad P(n)>n^{3/5 - \varepsilon}. \] In order to prove these results, the authors adapt inter alia a result of \textit{E. Bombieri, J. B. Friedlander} and \textit{H. Iwaniec} [Math. Ann. 277, 361--393 (1987; Zbl 0625.10036)].
scientific article; zbMATH DE number 5620966
scientific article