On the number of \(y\)-smooth natural numbers \(\leq x\) representable as a sum of two integer squares
Let \(b(n) = 1\) if \(n\) can be represented as a sum of two squares of integers and let \(b(n) = 0\) otherwise. The author studies the function \(B(x,y) = \sum_{n \leq x, P(n) \leq y} b(n)\), where \(P(n)\) denotes the largest prime factor of \(n\). For any given \(\varepsilon\) with \(0 < \varepsilon < 1/2\), he proves \[ B(x,x^{1/u}) = c\cdot {x\over \sqrt{\log x}} \sigma(u)(1 + o(1)), \quad x \to \infty, \] uniformly in \(1 \leq u \leq (\textstyle{1\over 2} - \varepsilon) \log \log x/\log \log \log x\). Here, \(\sigma(u)\) satisfies a differential-difference equation similar to the one satisfied by the Dickman function and \(c\) is a positive constant.
scientific article; zbMATH DE number 492800