$L^1$ 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 $$\int_{0}^{1} \Big | \sum_{n \leq N} f(n)e^{2πi n α} \Big | d α\leq N^{o(1)}$$ as $N \rightarrow \infty$, then there exists a quadratic character $χ$ such that for every $δ> 0$ the (logarithmic) proportion of primes $p \leq N$ such that $|f(p) - χ(p)| < δ$ tends to $1$ as $N \rightarrow \infty$. More generally we show that for all $N, Δ\geq 1$ and $1$-bounded multiplicative functions $f$, if $$\int_{0}^{1} \Big | \sum_{n \leq N} f(n) e^{2πi n α} \Big | d α\leq Δ$$ and the $L^{2}$ norm of $f$ over $[1, N]$ is $\geq N / 100$, then $f$ pretends to be a multiplicative character of conductor $\leq Δ^{2}$ on primes in $[Δ^{2}, N]$. We highlight that the result is uniform in $f$, $N$ and $Δ$ and sharp as far as the size of the conductor goes. Moreover, the restriction to primes $p \in [Δ^{2}, N]$ 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.