Multiplicative arithmetic function

An arithmetic function $f$ is multiplicative when $f(mn)=f(m)f(n)$ whenever $m$ and $n$ are coprime.

Multiplicativity means $f(mn)=f(m)f(n)$ whenever $m,n$ are coprime integers. It allows values of an arithmetic function to be assembled from its values on prime powers.

An arithmetic function $f$ is completely multiplicative when $f(mn)=f(m)f(n)$ for every pair of positive integers $m,n$, without a coprimality condition.

The divisor sum of an arithmetic function $f$ isIf $f$ is multiplicative, then $F$ is multiplicative as well.

The sum-of-divisors function isIt is the divisor sum of the identity arithmetic function $n\mapsto n$.

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

Let $X\ge 3$, and let $f$ be a normalized multiplicative arithmetic function supported on squarefree integers, with $|f(p)|\le 1$. Form the finite measure $\nu ={\sum }_{n}f(n)n^{-1-1/\log X}{\delta }_{\log n}$ using Dirac measures. Under the transform convention $\hat{\nu }(t)=\int e^{-itu}d\nu (u)$, its Euler product is ${\prod }_p(1+f(p)p^{-1-1/\log X-it})$. Expanding its logarithmic modulus, and replacing the damped prime weights by $1/p$ up to $X$ with Mertens first theorem, gives the displayed absolute bound, uniformly in real $t$. Hence $|\hat{\nu }(t)|\ge \delta \log X$ forces $D(f,n^{it};X{)}^2\le \log (C/\delta )$. The distance expression still defines a nonnegative pretentious score for prime values bounded by one; its actual metric interpretation uses unit-modulus values.

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 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 ${\sum }_{p\le x}(1+\cos (t\log p))/p$.