$Ω$-bounds for the partial sums of some modified Dirichlet characters II
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| Format: | Preprint |
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2025
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| _version_ | 1866912289972551680 |
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| author | Aymone, Marco Chaves, Ana Paula Ramos, Maria Eduarda |
| author_facet | Aymone, Marco Chaves, Ana Paula Ramos, Maria Eduarda |
| contents | 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 $1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_18228 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $Ω$-bounds for the partial sums of some modified Dirichlet characters II Aymone, Marco Chaves, Ana Paula Ramos, Maria Eduarda Number Theory 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 $1$. |
| title | $Ω$-bounds for the partial sums of some modified Dirichlet characters II |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.18228 |