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One quantitative form of Halász theorem is the following. If $f$ is multiplicative and $|f(n)|\le 1$, putThen, uniformly for $2\le T\le x$,Thus a bounded multiplicative arithmetic function can have a large mean only when it has small pretentious distance from some Archimedean character $n^{it}$.

Solved by gpt-5.6-sol high.

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.