On the sum of a prime and a \(k\)-free number (Q873797)

The author proves a refined asymptotic formula for the number of representations of a sufficiently large integer as a sum of a prime and a \(k\)-free number, \(k\geq 2\). This refinement of the Walfisz-Mirsky asymptotic formula [see \textit{L. Mirsky}, Am. Math. Mon. 56, 17--19 (1949; Zbl 0033.16203) and \textit{A. Walfisz}, Math. Z. 40, 592--607 (1935; Zbl 0013.10403)] depends on inserting a new term connected with the existence of the Siegel zero of Dirichlet \(L\)-functions and by sharpening the error term in the asymptotic formula. An analogous result, but with a weaker error term, can also be obtained via the circle method using some recent results on exponential sums over \(k\)-free numbers proved by \textit{J. Brüdern, A. Granville, A. Perelli, R. C. Vaughan} and \textit{T. D. Wooley} [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 356, No. 1738, 739--761 (1998; Zbl 0922.11071)].