| Title:
|
On the variation of certain fractional part sequences (English) |
| Title:
|
Sur la variation de certaines suites de parties fractionnaires (French) |
| Author:
|
Balazard, Michel |
| Author:
|
Benferhat, Leila |
| Author:
|
Bouderbala, Mihoub |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 (print) |
| ISSN:
|
2336-1298 (online) |
| Volume:
|
29 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
407-430 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $b>a>0$. We prove the following asymptotic formula $$ \sum _{n\geqslant 0} \big \lvert \{x/(n+a)\}-\{x/(n+b)\}\big \rvert =\frac {2}{\pi }\zeta (3/2)\sqrt {cx}+O(c^{2/9}x^{4/9})\,, $$ with $c=b-a$, uniformly for $x \geqslant 40 c^{-5}(1+b)^{27/2}$. (English) |
| Keyword:
|
Fractional part |
| Keyword:
|
Elementary methods |
| Keyword:
|
van der Corput estimates |
| MSC:
|
11N37 |
| idZBL:
|
Zbl 07484377 |
| idMR:
|
MR4355415 |
| . |
| Date available:
|
2022-01-10T10:04:37Z |
| Last updated:
|
2022-04-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149326 |
| . |
| Reference:
|
[1] Balazard, M.: Sur la variation totale de la suite des parties fractionnaires des quotients d'un nombre réel positif par les nombres entiers naturels consécutifs.Mosc. J. Comb. Number Theory, 7, 2017, 3-23, MR 3656668 |
| Reference:
|
[2] Corput, J.G. van der: Méthodes d'approximation dans le calcul du nombre des points ŕ coordonnées entičres.Enseign. Math., 23, 1923, 5-29, |
| Reference:
|
[3] Corput, J.G. van der: Neue zahlentheoretische Abschätzungen.Math. Ann., 89, 1923, 215-254, 10.1007/BF01455979 |
| Reference:
|
[4] Corput, J.G. van der: Zahlentheoretische Abschätzungen mit Anwendung auf Gitterpunktprobleme.Math. Z., 17, 1923, 250-259, 10.1007/BF01504346 |
| Reference:
|
[5] Wintner, A.: Square root estimates of arithmetical sum functions.Duke Math. J., 13, 1946, 185-193, 10.1215/S0012-7094-46-01319-1 |
| . |
