Moments of the shifted prime divisor function
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912403734659072 |
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| author | Gabdullin, Mikhail R. |
| author_facet | Gabdullin, Mikhail R. |
| contents | Let $ω^*(n) = \{d|n: d=p-1, \mbox{$p$ is a prime}\}$. We show that, for each integer $k\geq2$, $$ \sum_{n\leq x}ω^*(n)^k \asymp x(\log x)^{2^k-k-1}, $$ where the implied constant may depend on $k$ only. This confirms a recent conjecture of Fan and Pomerance. Our proof uses a combinatorial identity for the least common multiple, viewed as a multiplicative analogue of the inclusion-exclusion principle, along with analytic tools from number theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24050 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moments of the shifted prime divisor function Gabdullin, Mikhail R. Number Theory Let $ω^*(n) = \{d|n: d=p-1, \mbox{$p$ is a prime}\}$. We show that, for each integer $k\geq2$, $$ \sum_{n\leq x}ω^*(n)^k \asymp x(\log x)^{2^k-k-1}, $$ where the implied constant may depend on $k$ only. This confirms a recent conjecture of Fan and Pomerance. Our proof uses a combinatorial identity for the least common multiple, viewed as a multiplicative analogue of the inclusion-exclusion principle, along with analytic tools from number theory. |
| title | Moments of the shifted prime divisor function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2505.24050 |