Pretentious distance ($\mathbb{D}(f,g;x)$)

For multiplicative functions bounded by one, the pretentious distance up to $x$ is defined byIt measures how similarly $f$ and $g$ behave on primes.

The pretentious distance obeys

An Archimedean character on the positive integers is the completely multiplicative function $n\mapsto n^{it}$ for a real parameter $t$.

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.

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