$\mathrm{\Omega }$-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 $p\in S$, $\mathrm{\mid }f(p)\mathrm{\mid }\le 1$. If $\chi$ is primitive and for each $p\in S$ we have $\mathrm{\mid }f(p)\mathrm{\mid }=1$, we prove that ${\sum }_{n\le x}f(n)=\mathrm{\Omega }((\log x{)}^{(\mathrm{\mid }S\mathrm{\mid }-3)\mathrm{/}2})$. This makes progress on a Conjecture due to Klurman, Mangerel, Pohoata and Teräväinen, 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 $1$.