$\Omega$-bounds for the partial sums of some modified Dirichlet characters II

A modified Dirichlet character $f$ is a completely multiplicative function such that for some Dirichlet character $\chi$, $f(p)=\chi(p)$ for all but a finite number of primes $p\in S$, and for those exceptional primes $p\in S$, $|f(p)|\leq 1$. If $\chi$ is primitive and for each $p\in S$ we have $|f(p)|=1$, we prove that $\sum_{n\leq x}f(n)=\Omega((\log x)^{(|S|-3)/2})$. This makes progress on a Conjecture due to Klurman, Mangerel, Pohoata and Ter"av"ainen, c.f. Trans. Amer. Math. Soc., 374 (2021), pp. 7967--7990. Our proof combines tools from Analytic Number Theory, Harmonic Analysis, Baker's Theory on linear forms in logarithms and Discrepancy bounds for sequences uniformly distributed modulo $1$.