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PreprintarXiv licencearXiv

Few small primes suffice to cover unit groups

  • Scott D. Hughes
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Abstract

Klurman, Shparlinski and Teräväinen showed that there is a set of at most (log Q)1+ε primes of polylogarithmic size whose subset products cover the unit group (ℤ/qℤ)× for almost all moduli q≤ Q. We show that (1/log2+o(1))log Q primes suffice, which is optimal to first order, since m primes have at most 2m subset products. The proof combines a doubling argument of Erdős--Rényi type for random subset products in a finite abelian group, modified to tolerate a small set of exceptional characters, with a one-time ``repair'' of the few primitive characters whose L-functions have zeros near s=1. An exact weighted repair gives a variant with at most (2/log2+o(1))log Q primes and an exceptional set described explicitly.

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Cite

BibTeXRIS
BibTeX
@misc{hughes2026small,
  title = {{Few small primes suffice to cover unit groups}},
  author = {Scott D. Hughes},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04678},
  eprint = {2610.04678},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04678},
}
RIS
TY  - UNPB
TI  - Few small primes suffice to cover unit groups
AU  - Scott D. Hughes
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04678
UR  - https://arxiv.org/abs/2610.04678
AB  - Klurman, Shparlinski and Teräväinen showed that there is a set of at most (log Q)^1+ε primes of polylogarithmic size whose subset products cover the unit group (ℤ/qℤ)^× for almost all moduli q≤ Q. We show that (1/log2+o(1))log Q primes suffice, which is optimal to first order, since m primes have at most 2^m subset products. The proof combines a doubling argument of Erdős--Rényi type for random subset products in a finite abelian group, modified to tolerate a small set of exceptional characters, with a one-time ``repair'' of the few primitive characters whose L-functions have zeros near s=1. An exact weighted repair gives a variant with at most (2/log2+o(1))log Q primes and an exceptional set described explicitly.
ER  -