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

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.