Somewhat smooth numbers in short intervals
We say that a natural number \(n\) is \(y\)-smooth if all of its prime factors are less than or equal to \(y\). Let \(\Psi(x, y)\) be the number of \(y\)-smooth numbers \(n\) not exceeding \(x\). Focusing on the case \(y=x^a\), in the paper under review, the author investigates existence of \(x^a\)-smooth numbers in short intervals, for the case \(a>1/2\). Letting \(\psi(t)=t-[t]-1/2\), it is known that \[ \sum_{N\leq n\leq 2N}\psi\left(\frac{x}{n}\right)\ll\min\left(x^\theta, x^{k/(k+1)}N^{(l-k)/(k+1)}\right)\qquad(*) \] holds for any \(1\leq N\leq\sqrt{x}\) and any exponent pair \((k,l)\), where the best known value for \(\theta\) is \(\theta=517/1648+o(1)\approxeq 0.3137\). For any exponent pair \((k,l)\) and \(\theta\) as in Eq. \((*)\), and for any \(a\in(1/2, 1]\), the author shows that there exists constant \(K\) such that for \(x\geq z\geq Kx^{\min(\theta,b)}\), \[ \Psi(x,x^a)-\Psi(x-z,x^a)\gg\frac{z}{(\log x)^v}, \] where \(b=b(a,k,l)=(l+a(k-l))/(k+1)\) and \(v=\min_{u>1}(2^u-1)/(u-1)\approxeq 2.9882\). Based on a result due to \textit{D. R. Heath-Brown} [Proc. Steklov Inst. Math. 296 (1), 88--103 (2017; Zbl 1461.11110)], providing an exponential pair, the author proves that for each \([1/2,1)\), the above conclusion holds for some \(b<1-a-a(1-a)^3-4.32a(1-a)^5\). Finally, he gives an application to the so-called practical numbers, by proving that if \(\mathcal{A}\) is the set of positive integers containing \(n=1\) and all those \(n\geq 2\) with prime factorization \(n=p_1^{\alpha_1}\cdots p_k^{\alpha_k}\) with \(p_1=2<p_2<\cdots<p_k\) and satisfying \(p_i\leq p_1^{\alpha_1}\cdots p_{i-1}^{\alpha_{i-1}}\) for \(2\leq i\leq k\), then the interval \([x-x^{0.4872},x]\) contains at least \(x^{0.4872}/(\log x)^{-9.557}\) members of \(\mathcal{A}\), for all sufficiently large \(x\).
scientific article; zbMATH DE number 7659060
scientific article