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Improved explicit Burgess-strength bounds for composite moduli
Abstract
Consider the character sum S(M,N)=∑n=M+1M+Nχ(n) for a non-principal Dirichlet character χ(n) to a (possibly composite) modulus q. Burgess proved that S(M,N)≪ N1/2q3/16+ε, for any ε>0. This was made explicit by Jain-Sharma et al., but only for moduli q≥ exp(exp(9.594))≈ 8.03× 106373. We prove a stronger result that holds when q≥ 1.627× 1019.
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BibTeX
@misc{punch2026improved,
title = {{Improved explicit Burgess-strength bounds for composite moduli}},
author = {James Punch},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22939},
eprint = {2610.22939},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22939},
}
RIS
TY - UNPB
TI - Improved explicit Burgess-strength bounds for composite moduli
AU - James Punch
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22939
UR - https://arxiv.org/abs/2610.22939
AB - Consider the character sum S(M,N)=∑_n=M+1^M+Nχ(n) for a non-principal Dirichlet character χ(n) to a (possibly composite) modulus q. Burgess proved that S(M,N)≪ N^1/2q^3/16+ε, for any ε>0. This was made explicit by Jain-Sharma et al., but only for moduli q≥ exp(exp(9.594))≈ 8.03× 10^6373. We prove a stronger result that holds when q≥ 1.627× 10^19.
ER -