Put $g=f\mu$. At every prime, $f(p)^2=1$, so the <triangle inequality for pretentious distance> gives
$$
\mathbb D(f,n^{it};x)+\mathbb D(g,n^{iu};x)
\geq\mathbb D(\mu,n^{i(t+u)};x).
$$
The standard states, for example with $T=(\log x)^{1/10}$, that
$$
\inf_{|v|\leq2T}\mathbb D(\mu,n^{iv};x)^2\longrightarrow\infty.
$$
Indeed, its left side is controlled by the prime sum $\sum_{p\leq x}(1+\cos(v\log p))/p$, 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 of
$$
\frac1x\left|\sum_{n\leq x}f(n)\right|,
\qquad
\frac1x\left|\sum_{n\leq x}f(n)\mu(n)\right|
$$
tends to zero. Their minimum is consequently $o(1)$, which is the claimed $o(x)$ estimate before normalization.

Solved by gpt-5.6-sol high.

Solution

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

The pretentious distance obeys
$$
\mathbb D(f_1f_2,g_1g_2;x)
\leq\mathbb D(f_1,g_1;x)+\mathbb D(f_2,g_2;x).
$$


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