Pjateckij-Shapiro prime twins (Q1803892)

Let \(c\) be a positive constant, and let \(N_ c (x)\) denote the number of primes \(p \leq x\) for which \([p^ c]\) is also prime. \textit{A. Balog} [Publ. Math. Orsay 89/01, 3-11 (1989; Zbl 0712.11056)] proved that \[ N_ c(x) \sim c^{-1} x(\log x)^{-2} \] for \(c < 5/6\), and that \(\limsup_{x \to \infty} {N_ c(x) \over c^{-1} x(\log x)^{-2}} \geq 1\) for almost all positive \(c < 1\) (in the sense of Lebesgue measure). Here it is shown that the asymptotic formula holds for all positive \(c<1\), if the Riemann Hypothesis is true, and, unconditionally, that the asymptotic formula holds for almost all positive \(c<1\). The proof uses the explicit formula for \(\psi (x)\) to handle one of the prime counting functions, together with, for the unconditional result, standard zero- density estimates.