The structure of multiplicative functions with small partial sums

The link is currently to the submitted version. The published version will appear very soon.

The structure of multiplicative functions with small partial sums, Discrete Analysis 2020:6, 19 pp.

A multiplicative function is a function from $\mathbb N$ to $\mathbb C$ with the property that $f(ab)=f(a)f(b)$ whenever $a$ and $b$ are coprime. Multiplicative functions arise very naturally and are of fundamental importance in number theory. From the definition (and the fundamental theorem of arithmetic) it follows that a multiplicative function is determined by the values it takes on prime powers.

A particularly important example is the Möbius function $\mu$. This takes the value 1 at 1, -1 at all primes $p$, and 0 at all other prime powers. From this it follows that $\mu(n)$ is zero if $n$ has a repeated prime factor and is otherwise 1 if the number of prime factors of $n$ is even and -1 if the number of prime factors is odd.

An indication of the importance of $\mu$ is that the Riemann hypothesis is equivalent to the assertion that $|\sum_{n\leq x}\mu(n)|=x^{\frac 12+o(1)}$, or in other words, that the partial sum up to $x$ does not get much larger than $\sqrt x$ in magnitude. More generally, understanding the behaviour of partial sums of multiplicative functions pays significant dividends in number theory.

A notable feature of the Möbius function is that it takes the same value at each prime. While many of the most commonly studied multiplicative functions do not have this property, they frequently have the weaker property that their average along primes tends to a limit. It is therefore natural to ask what relationship there is between this limiting average, if it exists, and the partial sums.

This question was addressed by Landau in 1909, and his work was developed by Selberg and Delange. The resulting Landau-Selberg-Delange method (often called the LSD method) provides an asymptotic formula for the partial sums in terms of its average along primes. From the asymptotic formula, it can be deduced that the partial sums are small (that is, significantly smaller than $x$) only if the average is a non-positive integer.

For a multiplicative function to have small partial sums, it is not necessary for its average values along primes to converge. Consider for example the function $f(n)=\mu(n)n^{i\gamma}$ for some real number $\gamma$, which takes the value $-p^{i\gamma}$ at a prime $p$. The function $n^{i\gamma}=e^{i\gamma\log n}$ progresses round the unit circle at ever-decreasing speed, and the Möbius function, which behaves in a “random” way, provides a great deal of cancellation, so the partial sums of $f$ are still small. But because of the slow growth rate of $\log n$, the average of $f(p)$ up to $x$ does not converge.

The purpose of this paper is to look at the considerably harder inverse problem that arises from observations like this: suppose we know that the partial sums of a multiplicative function (satisfying suitable conditions) are small. What can we say about its behaviour along the primes? The result is that, provided the multiplicative function satisfies a certain not very stringent condition, then we must have on average that

$$f(p)\approx -p^{i\gamma_1}-p^{i\gamma_2}-\dots-p^{i\gamma_k}$$

for some real numbers $\gamma_1,\dots,\gamma_k$. More precisely, we must have that

$$\lim_{x\to\infty} x^{-1}\sum_{p\leq x}(f(p)+p^{i\gamma_1}+p^{i\gamma_2}+\dots+p^{i\gamma_k})\log p=0.$$

Moreover, if $|f(p)|\leq D$ for every $p$, then $k$ can be chosen to be at most $D$. (Note that each $p^{i\gamma_j}$ has modulus 1, so this is on average necessary as well.)

It is reasonably straightforward to prove that if $f$ satisfies a condition like this, then its partial sums are small, so this gives a very nice characterization of such functions. Note that if we set $k=1$ and $\gamma_1=0$, then $f(p)$ averages -1, so $f$ is in some sense close to the Möbius function.

This paper generalizes work of the first author, who showed the result when $D=1$. That is, the conclusion holds with $k=0$ or $1$ if $|f(p)|\leq 1$ for every $p$.