PreprintCC BY 4.0arXiv
A recurrence in logarithmic Möbius sums with coprimality conditions via Euler--Maclaurin
Abstract
We derive a recurrence for logarithmically weighted Möbius sums by combining Möbius inversion with the Euler-Maclaurin summation formula. Consequently, starting from estimates of orders zero, one and two, we obtain polynomial approximations for these sums for every nonnegative integer exponent of log. We then extend these estimates to sums with coprimality conditions via a convolution identity and to sums weighted by the reciprocal of the sum-of-divisors function. We also study the nonnegativity of the latter sums for every positive integer logarithmic order, establishing that they never change sign.
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BibTeX
@misc{alterman2026recurrence,
title = {{A recurrence in logarithmic Möbius sums with coprimality conditions via Euler--Maclaurin}},
author = {Sebastian Zuniga Alterman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17720},
eprint = {2610.17720},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17720},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A recurrence in logarithmic Möbius sums with coprimality conditions via Euler--Maclaurin
AU - Sebastian Zuniga Alterman
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.17720
UR - https://arxiv.org/abs/2610.17720
AB - We derive a recurrence for logarithmically weighted Möbius sums by combining Möbius inversion with the Euler-Maclaurin summation formula. Consequently, starting from estimates of orders zero, one and two, we obtain polynomial approximations for these sums for every nonnegative integer exponent of log. We then extend these estimates to sums with coprimality conditions via a convolution identity and to sums weighted by the reciprocal of the sum-of-divisors function. We also study the nonnegativity of the latter sums for every positive integer logarithmic order, establishing that they never change sign.
ER -