The distribution of the number of prime divisors of sums \(a+b\)
Let \(A,B\subset [1,x]\) be sets of integers and let \(F(z)\) denote the frequency of those pairs \((a,b)\), \(a\in A\), \(b\in B\) that satisfy \[ \omega (a+b)-\log \log x<z\sqrt{\log \log x}, \] where \(\omega\) denotes the number of prime divisors. \textit{P. Erdős}, \textit{H. Maier} and \textit{A. Sárközy} [Trans. Am. Math. Soc. 302, 269--280 (1987; Zbl 0617.10038)] showed that for large sets \(F(z)\) approximates the standard normal distribution \(G(z)\). The authors prove the following effective generalization of this result. Assume that \[ \vert B\vert >x(\log x)^{-c}\quad \text{and}\quad \vert A\vert >x \exp (-\lambda (\log \log x)^{1/2} \log \log x); \] then we have \[ F(z)-G(z)\ll \lambda +(\log \log x)^{\mp\varepsilon}, \] where the implied constant depends on \(c\) and \(\varepsilon\).
scientific article; zbMATH DE number 4053748