More primes and polynomials (Q1417939)

Let \(f(x)\in \mathbb Q(x)\) have positive degree, and let \(r\in\mathbb Q_{>0}\). Then it is shown that there is a value \(k\in \{1,2,3\}\) such that the equation \[ (p_1+1)f(p_2)=r^k(p_3+1)f(p_4) \] has infinitely many solutions in primes \(p_i\). This improves a corresponding result for \(k\in\{1,2,3,4\}\) of the author [Invent. Math. 149, 453--487 (2002; Zbl 1041.11062)]. The result is achieved by examining the number \(r(n)\) of representations of an integer \(n\) as \(M^{-1}(p_1+1)f(p_2)^{-1}\), where \(M\) is a fixed highly composite number, and the primes \(p_1,p_2\) are in specified ranges. By taking \(p_2\) to be of order \((\log p_1)^\alpha\) for a suitable \(\alpha\) one is able to estimate \(\sum r(n)\) via the Siegel-Walfisz Theorem. One may also bound \(\sum r(n)^2\) from above using a sieve method. A simple approach shows that \(r(n) > 0\) for an asymptotic proportion at least 1/4 of all integers, and this is sufficient for the version with \(k\leq 4\). A more sophisticated sieve bound, using ideas of \textit{J. Chen} [Sci. Sin. 16, 157--176 (1973; Zbl 0319.10056)], allows one to show that \(r(n) > 0\) for a proportion strictly greater than 1/4 of integers \(n\), leading to the result stated.