Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 117 / 3 / e / Solution

Write $M_0=CR{T}^2$ and split the last sum in part d according as ${\tau }_4(m)<X$ or ${\tau }_4(m)\ge X$. For the first part, the reciprocal fractional-part sum near a rational givesFor the second part, use the supplied second-moment estimate and truncation of a divisor weight by its second moment:Substitute $M_0\asymp R{T}^2$, take fourth roots, and factor out $R^{1/2}{T}^{1/2}$. The four terms from the bounded-weight estimate become, after harmless enlargement by $(\log q)(\log RT{)}^{O(1)}$,with the smaller $XR$ term absorbed by $X{M}_0$. The large-weight part contributes $X^{-1/4}$. Finally, the two diagonal terms from part d contribute $R^{-1/4}$ and $T^{-1/4}$; since $X\ge 1$, the first is absorbed by $X^{1/4}{R}^{-1/4}$. Therefore