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

TL;DR

This work proves that for a modification $f$ of a primitive Dirichlet character $\chi$ with modification set $S$, if $|f(p)|=1$ for all $p\in S$, then $\sum_{n\le x} f(n)=\Omega((\log x)^{(|S|-3)/2})$, improving our understanding of $\Omega$-bounds toward the Klurman--Mangerel--Pohoata--Teräväinen conjecture. The authors combine zero-free regions and functional equations for $L(s,\chi)$, a harmonic-analysis framework via Plancherel's identity, Baker's theory on linear forms in logarithms, and discrepancy bounds for sequences uniformly distributed modulo $1$ through a toral dynamical-systems viewpoint. The approach leverages a dynamical system on the torus, where $\boldsymbol\alpha=(\log p_1,\dots,\log p_{|S|})$ and phases $\boldsymbol\theta$ from $f(p)$, to force $E_f(s)$ to be small on strategically chosen sets, and then translates this into a lower bound on the partial sums. The result provides concrete, nontrivial growth rates in terms of $|S|$ and advances the conjectural picture for near-counterexamples to uniform-sign Dirichlet character sums, with potential implications for related discrepancy problems and the Erdős discrepancy framework.

Abstract

A modified Dirichlet character $f$ is a completely multiplicative function such that for some Dirichlet character $χ$, $f(p)=χ(p)$ for all but a finite number of primes $p\in S$, and for those exceptional primes $p\in S$, $|f(p)|\leq 1$. If $χ$ is primitive and for each $p\in S$ we have $|f(p)|=1$, we prove that $\sum_{n\leq x}f(n)=Ω((\log x)^{(|S|-3)/2})$. 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 $1$.

Theorems & Definitions (7)

• Theorem 1.1
• Theorem 1.2
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• proof : Proof of Theorem \ref{['Teorema principal']}