Some remarks on the [x/n]-sequence: Some remarks on the \([x/n]\)-sequence
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| Název: | Some remarks on the [x/n]-sequence: Some remarks on the \([x/n]\)-sequence |
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| Autoři: | Kota Saito, Yuta Suzuki, Wataru Takeda, Yuuya Yoshida |
| Zdroj: | Research in Number Theory. 11 |
| Publication Status: | Preprint |
| Informace o vydavateli: | Springer Science and Business Media LLC, 2025. |
| Rok vydání: | 2025 |
| Témata: | exponent pair, Primary: 11N37. Secondary: 11N25, 11N69, 11L03, 11L07, Number Theory, integral part, FOS: Mathematics, Asymptotic results on arithmetic functions, Distribution of integers in special residue classes, Number Theory (math.NT), Estimates on exponential sums, arithmetic functions, Trigonometric and exponential sums (general theory), Distribution of integers with specified multiplicative constraints, exponential sums |
| Popis: | After the work of Bordellès, Dai, Heyman, Pan and Shparlinki (2018) and Heyman (2019), several authors studied the averages of arithmetic functions over the sequence $[x/n]$ and the integers of the form $[x/n]$. In this paper, we give three remarks on this topic. Firstly, we improve the result of Wu and Yu (2022) on the distribution of the integers of the form $[x/n]$ in arithmetic progressions by using a variant of Dirichlet's hyperbola method. Secondly, we prove an asymptotic formula for the number of primitive lattice points with coordinates of the form $[x/n]$, for which we introduce a certain averaging trick. Thirdly, we study a certain "multiplicative" analog of the Titchmarsh divisor problem. We derive asymptotic formulas for such "multiplicative" Titchmarsh divisor problems for "small" arithmetic functions and the Euler totient function with the von Mangoldt function. However, it turns out that the average of the Euler totient function over the $[x/p]$-sequence seems rather difficult and we propose a hypothetical asymptotic formula for this average. 27 pages |
| Druh dokumentu: | Article |
| Popis souboru: | application/xml |
| Jazyk: | English |
| ISSN: | 2363-9555 2522-0160 |
| DOI: | 10.1007/s40993-025-00632-y |
| DOI: | 10.48550/arxiv.2312.15642 |
| Přístupová URL adresa: | http://arxiv.org/abs/2312.15642 https://zbmath.org/8034693 https://doi.org/10.1007/s40993-025-00632-y |
| Rights: | Springer Nature TDM arXiv Non-Exclusive Distribution |
| Přístupové číslo: | edsair.doi.dedup.....c2da739e6f79935c176bbcce1de69971 |
| Databáze: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstrakt: | After the work of Bordellès, Dai, Heyman, Pan and Shparlinki (2018) and Heyman (2019), several authors studied the averages of arithmetic functions over the sequence $[x/n]$ and the integers of the form $[x/n]$. In this paper, we give three remarks on this topic. Firstly, we improve the result of Wu and Yu (2022) on the distribution of the integers of the form $[x/n]$ in arithmetic progressions by using a variant of Dirichlet's hyperbola method. Secondly, we prove an asymptotic formula for the number of primitive lattice points with coordinates of the form $[x/n]$, for which we introduce a certain averaging trick. Thirdly, we study a certain "multiplicative" analog of the Titchmarsh divisor problem. We derive asymptotic formulas for such "multiplicative" Titchmarsh divisor problems for "small" arithmetic functions and the Euler totient function with the von Mangoldt function. However, it turns out that the average of the Euler totient function over the $[x/p]$-sequence seems rather difficult and we propose a hypothetical asymptotic formula for this average.<br />27 pages |
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| ISSN: | 23639555 25220160 |
| DOI: | 10.1007/s40993-025-00632-y |