Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 150 / 1 / b / Solution

The Dirichlet hyperbola method counts each factorization $n=ab$ once and gives, with $y=\lfloor \sqrt{x}\rfloor$,Using $\lfloor x/a\rfloor =x/a+O(1)$ and the harmonic number estimate $H_y=\log y+\gamma +O(1/y)$, where $\gamma$ is the Euler--Mascheroni constant, we obtainThus one may take $C=2\gamma -1$.

Solved by gpt-5.6-sol high.