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A power-saving asymptotic for a Möbius-weighted GCD quadratic form
Abstract
For the Möbius-weighted quadratic form T(N)=∑m,n≤ Nμ(m)μ(n)(m,n)2/(mn), we prove that T(N)=cN+Oε(N2/3+ε) for every ε>0, where c>0 is given by an absolutely convergent series of finite Möbius sums multiplied by a convergent Euler product. The proof combines the classical Jordan-totient decomposition of the GCD kernel with a mean-value estimate that is uniform in a coprimality parameter. Grouping the resulting squares according to ⌊ N/d⌋ gives the error term and a nonnegative series for c.
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BibTeX
@misc{azimifard2026powersaving,
title = {{A power-saving asymptotic for a Möbius-weighted GCD quadratic form}},
author = {Ahmadreza Azimifard},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00525},
eprint = {2611.00525},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00525},
}
RIS
TY - UNPB
TI - A power-saving asymptotic for a Möbius-weighted GCD quadratic form
AU - Ahmadreza Azimifard
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00525
UR - https://arxiv.org/abs/2611.00525
AB - For the Möbius-weighted quadratic form T(N)=∑_m,n≤ Nμ(m)μ(n)(m,n)^2/(mn), we prove that T(N)=cN+O_ε(N^2/3+ε) for every ε>0, where c>0 is given by an absolutely convergent series of finite Möbius sums multiplied by a convergent Euler product. The proof combines the classical Jordan-totient decomposition of the GCD kernel with a mean-value estimate that is uniform in a coprimality parameter. Grouping the resulting squares according to ⌊ N/d⌋ gives the error term and a nonnegative series for c.
ER -