Supercharacter theory and applications to Ramanujan sums over a finite Frobenius ring
Abstract
The theory of Ramanujan sums has been playing a fundamental role in several subfields of mathematics. They appear in the theory of special values of zeta functions, spectral graph theory, representation theory, and analytic number theory. Recent work has shown that classical Ramanujan sums can also be interpreted as a super-Fourier transform via the theory of supercharacters for the ring ℤ/n. In this article, building upon our recent work on supercharacters over an arbitrary finite Frobenius ring, we explore additional arithmetical properties of Ramanujan sums. Our approach provides a unified framework that generalizes various results in the literature regarding these sums. As a by-product, we also describe a new criterion for determining when a finite commutative ring is Frobenius, which could be of independent interest to the algebra community.
Related
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BibTeX
@misc{nguyen2026supercharacter,
title = {{Supercharacter theory and applications to Ramanujan sums over a finite Frobenius ring}},
author = {Tung T. Nguyen and Nguyen Duy Tân and Enrique Treviño},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30407},
eprint = {2609.30407},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30407},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Supercharacter theory and applications to Ramanujan sums over a finite Frobenius ring
AU - Tung T. Nguyen
AU - Nguyen Duy Tân
AU - Enrique Treviño
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30407
UR - https://arxiv.org/abs/2609.30407
AB - The theory of Ramanujan sums has been playing a fundamental role in several subfields of mathematics. They appear in the theory of special values of zeta functions, spectral graph theory, representation theory, and analytic number theory. Recent work has shown that classical Ramanujan sums can also be interpreted as a super-Fourier transform via the theory of supercharacters for the ring ℤ/n. In this article, building upon our recent work on supercharacters over an arbitrary finite Frobenius ring, we explore additional arithmetical properties of Ramanujan sums. Our approach provides a unified framework that generalizes various results in the literature regarding these sums. As a by-product, we also describe a new criterion for determining when a finite commutative ring is Frobenius, which could be of independent interest to the algebra community.
ER -