Few small primes suffice to cover unit groups

Scott D. Hughes

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arXiv · 2610.04678

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 $(\log Q)^{1+\varepsilon}$ primes of polylogarithmic size whose subset products cover the unit group $(\mathbb{Z}/q\mathbb{Z})^\times$ for almost all moduli $q\le Q$. 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.

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BibTeX RIS

Scott D. Hughes. 2026-10-03. Few small primes suffice to cover unit groups. https://arxiv.org/abs/2610.04678

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