Remarks on the differences between consecutive primes (Q1140115)

Define \(F(n,k)\), to be the number of solution of \(p_j-p_i=2k\), \((p_j\leq n)\), and let \(f(n,k)\) be the number for which \(j=i+1\). This paper is concerned with the behaviour as \(n\to\infty\) of the maximum values of \(f(n,k)\) and \(F(n,k)\), and with the least values (\(k_n\) and \(K_n\) respectively) for which the maxima are attained. Hardy and Littlewood gave a conjectured asymptotic formula for \(F(n,k)\), for fixed \(k\). On the assumption of this it is shown that \[ f(n,k_n)/\{n(\log n)^{-2}\}\to\infty \] and that \(k_n\to\infty\). In contrast it is shown that \[ F(n,K_n)/\{n(\log\log n)(\log n)^{-2}\}\gg 1 \] without any hypothesis.