On the exceptional set of Hardy-Littlewood's numbers in short intervals (Q1769864)

Define a HL-number as being an integer which is the sum of a prime and a \(k\)th power of an integer: Hardy and Littlewood conjectured that every sufficiently large integer is either a \(k\)th power or is an HL-number. Let \(E_ k\) denote the set of ``exceptional'' positive integers which are not of either of these forms, and let \(E_ k(X,X+H)\) denote the number of exceptional integers in the interval \([X,X+H]\). Abbreviate \(E_ k(1,X)\) to \(E_ k(X)\), so that the conjecture was \(E_ k(X) \ll 1\). \textit{A. I. Vinogradov} [Acta Arith. 46, 33--56 (1985; Zbl 0597.10048)] and independently \textit{R. Brünner, A. Perelli} and \textit{J. Pintz} [Acta Math. Hung. 53, No. 3/4, 347--365 (1989; Zbl 0683.10039)] showed that \(E_ 2(X) \ll X^ {1-\delta}\) for some \(\delta>0\). For the related short-interval problem, \textit{A. Perelli} and \textit{J. Pintz} [J. Number Theory 54, No. 2, 297--308 (1995; Zbl 0851.11056)] and independently \textit{H. Mikawa} [Tsukuba J. Math. 17, No. 2, 299--310 (1993; Zbl 0802.11037)] showed that \(E_ 2(X,H) \ll H\log^ {-A}\!X\) when \(H \geq X^ {7/24+\varepsilon}\). These results were extended to \(E_ k\) for \(k \geq 2\) by \textit{A. Zaccagnini} [Mathematika 39, No. 2, 400--421 (1992; Zbl 0760.11026)] and by \textit{A. Perelli} and \textit{A. Zaccagnini} [Izv. Ross. Akad. Nauk, Ser. Mat. 59, No. 1, 185--200 (1995; Zbl 0996.11065)] respectively. The author makes a quantitative improvement in showing that \(E_ k(X,H) \ll H^ {1-\delta/(5K)}\), in which \(K=2^ {k-2}\), when \(\delta>0\) and \(H \geq X^ {7/12(1-1/k)+\delta}\). As in the earlier papers on this topic, the treatment uses the circle method, with a refined version of the input from density estimates for the zeros of Dirichlet \(L\)-functions.