Mean values of multiplicative functions and applications to residue-class distribution
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909734067503104 |
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| author | Pollack, Paul Roy, Akash Singha |
| author_facet | Pollack, Paul Roy, Akash Singha |
| contents | We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi-Erdős function $A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo $q$, where $q$ varies uniformly up to a fixed power of $\log x$. We establish a similar result for the equidistribution of the Euler totient function $ϕ(n)$ among the coprime residues to the "correct" moduli $q$ that vary uniformly in a similar range, and also quantify the failure of equidistribution of the values of $ϕ(n)$ among the coprime residue classes to the "incorrect" moduli. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16266 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mean values of multiplicative functions and applications to residue-class distribution Pollack, Paul Roy, Akash Singha Number Theory Primary 11A25, Secondary 11N36, 11N64 We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi-Erdős function $A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo $q$, where $q$ varies uniformly up to a fixed power of $\log x$. We establish a similar result for the equidistribution of the Euler totient function $ϕ(n)$ among the coprime residues to the "correct" moduli $q$ that vary uniformly in a similar range, and also quantify the failure of equidistribution of the values of $ϕ(n)$ among the coprime residue classes to the "incorrect" moduli. |
| title | Mean values of multiplicative functions and applications to residue-class distribution |
| topic | Number Theory Primary 11A25, Secondary 11N36, 11N64 |
| url | https://arxiv.org/abs/2402.16266 |