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Halász's theorem bounds the mean of a bounded multiplicative arithmetic function in terms of its least pretentious distance from an Archimedean character. A large mean can occur only when this distance is small.

For arithmetic functions $f,g$ bounded in modulus by one, their pretentious distance up to $x$ is defined by

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 $n^{it}$.

Because $\mu (p)=-1$, the squared pretentious distance from one to the Möbius function is twice the reciprocal-prime sum. The Mertens second theorem therefore gives $D(1,\mu ;X{)}^2=2\log \log X+2B_1+o(1)$. In particular the distance diverges, illustrating that a multiplicative function can be far from the constant function even though both have unit-modulus prime values.

For unit-modulus prime values, pretentious distance is the Euclidean metric on the finite vectors $(f(p)/\sqrt{2p}{)}_{p\le X}$. On whole arithmetic functions it is only a pseudometric: it sees neither values on prime powers nor primes above the cutoff. Even for every cutoff, the functions one and ${\mu }^2$ have distance zero but differ on nonsquarefree integers. The norm representation proves the triangle inequality and identifies exactly the quotient on which separation holds.

The Möbius function stays far in pretentious distance from every Archimedean character. One useful uniform form isThis follows by estimating the prime sum ${\sum }_{p\le x}(1+\cos (t\log p))/p$.