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

Let $f$ be a real-valued $1$-bounded multiplicative function. Suppose that the mean-value of $f^2$ exists, and $${\int }_0^1\mid \sum _{n\le N}f(n)e^{2\pi in\alpha }\mid d\alpha \le N^{o(1)}$$ as $N\to \mathrm{\infty }$, then there exists a quadratic character $\chi$ such that for every $\delta >0$ the (logarithmic) proportion of primes $p\le N$ such that $\mathrm{\mid }f(p)-\chi (p)\mathrm{\mid }<\delta$ tends to $1$ as $N\to \mathrm{\infty }$. More generally we show that for all $N,\mathrm{\Delta }\ge 1$ and $1$-bounded multiplicative functions $f$, if $${\int }_0^1\mid \sum _{n\le N}f(n)e^{2\pi in\alpha }\mid d\alpha \le \mathrm{\Delta }$$ and the $L^2$ norm of $f$ over $[1,N]$ is $\ge N\mathrm{/}100$, then $f$ pretends to be a multiplicative character of conductor $\le {\mathrm{\Delta }}^2$ on primes in $[{\mathrm{\Delta }}^2,N]$. We highlight that the result is uniform in $f$, $N$ and $\mathrm{\Delta }$ and sharp as far as the size of the conductor goes. Moreover, the restriction to primes $p\in [{\mathrm{\Delta }}^2,N]$ turns out to be sharp in a suitably generalized version of this result, concerning sequences $f$ that are close $1\mathrm{\%}$ of the time to multiplicative functions.