Averaged forms of two conjectures of Erdős and Pomerance, and their applications (Q2089036)

Letting \(P^+(n)\) be the largest prime factor of integer \(n\), a challenging conjecture due to \textit{P. Erdős} and \textit{C. Pomerance} [Aequationes Math. 17, 311--321 (1978; Zbl 0379.10027)] asserts that the average of the number of integers \(n\leq x\) satisfying \(P^+(n)\leq x^s\) and \(P^+(n+1)\leq x^t\) with \(s,t\in(0,1)\), as \(x\to\infty\), tends to \(\rho(1/s)\rho(1/t)\), where \(\rho\) is the so-called Dickman function. In the paper under review, the authors study this conjecture on average by approximating the following quantity for certain values of parameters \[ \sum_{|h|\leq H}\Big|\sum_{\substack{x<n\leq 2x\\ P^+(n)\leq y_1 \\ P^+(n+h)\leq y_2}}1-\rho(u_1)\rho(u_2)x\Big|. \] The authors also provide a similar study on average concerning another conjecture due to Erdős-Turán asserting that the asymptotic density of integers \(n\) with \(P^+(n)<P^+(n+1)\) exists and equals \(1/2\). For that, they approximate the following quantity for certain values of parameters \[ \sum_{|h|\leq H}\Big|\sum_{\substack{x<n\leq 2x\\ P^+(n)<P^+(n+h)}}1-\frac{x}{2}\Big|. \] As an application to their results, related by the \(abc\)-conjecture, the authors study portion of the triples of integers \((a, b, c)\) such that \(a+b=c\) and \(c\ll P^+(a)P^+(c)\).