On the variation of certain fractional part sequences

Balazard, Michel, Benferhat, Leila, and Bouderbala, Mihoub. "On the variation of certain fractional part sequences." Communications in Mathematics 29.3 (2021): 407-430. <http://eudml.org/doc/298187>.

@article{Balazard2021,
abstract = {Let $b>a>0$. We prove the following asymptotic formula \[ \sum \_\{n\geqslant 0\} \big \vert \lbrace x/(n+a)\rbrace -\lbrace x/(n+b)\rbrace \big \vert =\frac\{2\}\{\pi \}\zeta (3/2)\sqrt\{cx\}+O(c^\{2/9\}x^\{4/9\})\,, \] with $c=b-a$, uniformly for $x \geqslant 40 c^\{-5\}(1+b)^\{27/2\}$.},
author = {Balazard, Michel, Benferhat, Leila, Bouderbala, Mihoub},
journal = {Communications in Mathematics},
keywords = {Fractional part; Elementary methods; van der Corput estimates},
language = {eng},
number = {3},
pages = {407-430},
publisher = {University of Ostrava},
title = {On the variation of certain fractional part sequences},
url = {http://eudml.org/doc/298187},
volume = {29},
year = {2021},
}

TY - JOUR
AU - Balazard, Michel
AU - Benferhat, Leila
AU - Bouderbala, Mihoub
TI - On the variation of certain fractional part sequences
JO - Communications in Mathematics
PY - 2021
PB - University of Ostrava
VL - 29
IS - 3
SP - 407
EP - 430
AB - Let $b>a>0$. We prove the following asymptotic formula \[ \sum _{n\geqslant 0} \big \vert \lbrace x/(n+a)\rbrace -\lbrace x/(n+b)\rbrace \big \vert =\frac{2}{\pi }\zeta (3/2)\sqrt{cx}+O(c^{2/9}x^{4/9})\,, \] with $c=b-a$, uniformly for $x \geqslant 40 c^{-5}(1+b)^{27/2}$.
LA - eng
KW - Fractional part; Elementary methods; van der Corput estimates
UR - http://eudml.org/doc/298187
ER -