$L^1$ means of exponential sums with multiplicative coefficients. II

TL;DR

This paper characterizes multiplicative functions with small exponential sums, showing they closely resemble quadratic characters on primes, with results that are uniform, sharp, and extend to sequences near multiplicative functions.

Contribution

It establishes a uniform, sharp characterization of multiplicative functions with small exponential sums, linking them to quadratic characters and extending to near-multiplicative sequences.

Findings

Functions with small exponential sums resemble quadratic characters on primes.

The results are uniform in the function, N, and Δ, and optimal regarding the conductor size.

The prime restriction range is shown to be sharp in generalized settings.

Abstract

Let $f$ be a real-valued $1$-bounded multiplicative function. Suppose that the mean-value of $f^{2}$ exists, and $$\int_{0}^{1} \Big | \sum_{n \leq N} f(n)e^{2\pi i n \alpha} \Big | d \alpha\leq N^{o(1)}$$ as $N \rightarrow \infty$, then there exists a quadratic character $\chi$ such that for every $\delta > 0$ the (logarithmic) proportion of primes $p \leq N$ such that $|f(p) - \chi(p)| < \delta$ tends to $1$ as $N \rightarrow \infty$. More generally we show that for all $N, \Delta \geq 1$ and $1$-bounded multiplicative functions $f$, if $$\int_{0}^{1} \Big | \sum_{n \leq N} f(n) e^{2\pi i n \alpha} \Big | d \alpha \leq \Delta$$ and the $L^{2}$ norm of $f$ over $[1, N]$ is $\geq N / 100$, then $f$ pretends to be a multiplicative character of conductor $\leq \Delta^{2}$ on primes in $[\Delta^{2}, N]$. We highlight that the result is uniform in $f$, $N$ and $\Delta$ and sharp as far as the size of the conductor goes. Moreover, the restriction to primes $p \in [\Delta^{2}, N]$ turns out to be sharp in a suitably generalized version of this result, concerning sequences $f$ that are close $1\%$ of the time to multiplicative functions.