Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 47 min ago
PreprintarXiv licencearXiv

The least prime with a given primitive root and a variant of the larger sieve

  • Daniel R. Johnston
  • Sunil Naik
Open at sourcePDFDOICite

Abstract

For an integer g not equal to −1 or a square, we study the least prime pg such that g is a primitive root modulo pg. We show that, assuming the Generalised Riemann Hypothesis, all such g with |g|∈[N,2N] have pg≤(log |g|)3.44 with at most O((log N)2.44) exceptions. This can be directly compared to the uniform bound pg≤(log |g|)19 proven recently by Fan and Pollack. To obtain our result, we prove a new variant of Gallagher's larger sieve, which may be of independent interest. In addition to our conditional result, we discuss other ``almost all" bounds for pg that hold unconditionally.

Related

  1. Only finitely many modular Jacobians are supersingular modulo a given prime
    Aarya J. Kumar, Sargam Mondal, Erick Ross +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. On the sum of least prime factors in short intervals
    Xu Zhang
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. On a variant of the Piatetski-Shapiro prime number theorem
    Linji Long, Jinjiang Li, Min Zhang
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Exact curve counting of given word length on the once-punctured torus
    Filippo Baroni, David Fisac, Mingkun Liu
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsGeometry & topologyarXiv
  5. Counting Integral Matrices with Given Cubic Characteristic Polynomial
    Philipp Habegger
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{johnston2026least,
  title = {{The least prime with a given primitive root and a variant of the larger sieve}},
  author = {Daniel R. Johnston and Sunil Naik},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08569},
  eprint = {2610.08569},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08569},
}
RIS
TY  - UNPB
TI  - The least prime with a given primitive root and a variant of the larger sieve
AU  - Daniel R. Johnston
AU  - Sunil Naik
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08569
UR  - https://arxiv.org/abs/2610.08569
AB  - For an integer g not equal to −1 or a square, we study the least prime p_g such that g is a primitive root modulo p_g. We show that, assuming the Generalised Riemann Hypothesis, all such g with |g|∈[N,2N] have p_g≤(log |g|)^3.44 with at most O((log N)^2.44) exceptions. This can be directly compared to the uniform bound p_g≤(log |g|)^19 proven recently by Fan and Pollack. To obtain our result, we prove a new variant of Gallagher's larger sieve, which may be of independent interest. In addition to our conditional result, we discuss other ``almost all" bounds for p_g that hold unconditionally.
ER  -