Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 150 / 2 / e

Put $g=f\mu$. At every prime, $f(p{)}^2=1$, so the triangle inequality for pretentious distance givesThe standard strong aperiodicity of the Möbius function states, for example with $T=(\log x{)}^{1/10}$, thatIndeed, its left side is controlled by the prime sum ${\sum }_{p\le x}(1+\cos (v\log p))/p$, uniformly in this range.

Choose $t$ and $u$ minimizing the two distances in Halász theorem. The displayed triangle inequality implies that at least one of $M(f;x,T)$ and $M(g;x,T)$ tends to infinity. Halász's bound, and $T\to \infty$, then show that at least one oftends to zero. Their minimum is consequently $o(1)$, which is the claimed $o(x)$ estimate before normalization.

Solved by gpt-5.6-sol high.