Additive problems with smooth integers (Q2833602)

The set \(S(X,Y)\) of \(Y\)-smooth integers up to \(X\) is the set of positive integers \(\leq X\) that are free of prime factors \(>Y\).NEWLINENEWLINE In the paper under review the authors study the number \(R_s(n)\) of representations of a positive integer \(n\,(\leq X)\) in the form \(n=m^2+k\), where \(m\) is a natural number and \(k\in S(X,Y)\). They also discuss the number \(R_p(n)\) of representations of \(n\,(\leq X)\) in the form \(n=p+k\), where \(p\) is a prime number and \(k\in S(X,Y)\). Let NEWLINE\[NEWLINER^*_0={1\over 2} \sum_{\substack{ m+s=n\\ s<n,\,m<n}} \rho\Biggl({\log s\over\log Y}\Biggr)m^{-1/2}\text{ and }R^*_1= \sum_{\substack{ m+s=n\\ s<n,\,m<n}} \rho\Biggl({\log s\over\log Y}\Biggr)(\log m)^{-1},NEWLINE\]NEWLINE where \(\rho\) is the Dickman function. Let \(C_0>0\) be fixed, NEWLINE\[NEWLINEY\geq \exp(C_0(\log X)(\log\log\log X)/\log\log X),NEWLINE\]NEWLINE \(L>0\) arbitrarily large and \(\varepsilon> 0\). Using the circle method and recent results for exponential sums over smooth numbers, the authors prove that, for all \(n\leq X\) with at most \(O(X(\log X)^{-L})\) exceptions, the following estimates hold: NEWLINE\[NEWLINER_s(n)= R^*_0(1+ O((\log Y)^{-(1-\varepsilon)})),\;R_p(n)= R^*_1(1+ O((\log Y)^{-(1-\varepsilon)})),NEWLINE\]NEWLINE where the \(O\)-constants depend on \(C_0\), \(L\), \(\varepsilon\).