On the sum of a prime and a square
Let \(\nu(n)\) denote the number of representations of the natural number \(n\) as a sum of a prime and a square. \textit{G. H. Hardy} and \textit{J. E. Littlewood} [Acta Math. 44, 1-70 (1922; JFM 48.0143.04)] conjectured the asymptotic formula \(\nu(n)\sim {\mathfrak S}(n) \sqrt{n}/ \log n\) when \(n\) is not a perfect square, where \({\mathfrak S}(n)\) is an infinite product over the primes. The author proves that \[ \sum_ n |\nu(n)- {\mathfrak S} (n) \sqrt{n}/ \log n|^ 2 \ll_{\Theta,A} x^{1+\Theta} (\log x)^{- A}, \] where the sum is over \(n\in [x- x^ \Theta, x]\), and \(1/2< \Theta\leq 1\), thus improving a result of \textit{R. J. Miech} [Trans. Am. Math. Soc. 130, 494-512 (1968; Zbl 0164.35202)] which was the case \(\Theta=1\). Let \(E(x)\) denote the cardinality of the set of natural numbers \(n\leq x\) which have \(\nu(n) =0\). Improving on a result of \textit{A. I. Vinogradov} [Acta Arith. 46, 33-56 (1985; Zbl 0597.10048)], the author proves that \(E(x) \ll x^{1/2} (\log x)^ 5\) under the extended Riemann hypothesis and gives the unconditional upper bound \(E(x+ x^ \Theta)- E(x)\ll_{\Theta,A} x^ \Theta (\log x)^{-A}\), where \(7/24<\Theta \leq 1\) and \(A>0\). The sharpest unconditional upper bounds for \(E(x)\) can be found in Vinogradov's paper and in \textit{R. Brünner}, \textit{A. Perelli} and \textit{J. Pintz} [Acta Math. Hung. 53, 347-365 (1989; Zbl 0683.10039)].
scientific article; zbMATH DE number 591447