Extreme Values of Quadratic Dirichlet L-functions over Monic Irreducible Polynomials in 𝔽_q[t]

@misc{hossain2026extreme,
  title = {{Extreme Values of Quadratic Dirichlet $L$-functions over Monic Irreducible Polynomials in $\mathbbF_q[t]$}},
  author = {Ahammad Mostafa Hossain},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.18452},
  eprint = {2609.18452},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18452},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Extreme Values of Quadratic Dirichlet L-functions over Monic Irreducible Polynomials in 𝔽_q[t]
AU  - Ahammad Mostafa Hossain
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.18452
UR  - https://arxiv.org/abs/2609.18452
AB  - In this paper, we establish a new Ω-result for the values |L(σ,χ_P)| of quadratic Dirichlet L-functions at and near the central point, where σ=1/2+κ/(ln g), κ≥ 0, and P ranges over monic irreducible polynomials associated with hyperelliptic curves of genus g over a fixed finite field 𝔽_q. We consider the asymptotic setting in which q is fixed and g→∞. More precisely, for every ε∈ (0,1/2) andevery fixed κ≥ 0, we prove that max_P ∈ 𝒫_2g+1 |L(σ,χ_P)| ≫ exp ((e^−κ ln q √((√q+1)/(√q−1)(1/2−ε)) +o(1))√((gln_2 g)/(ln g)) ), where 𝒫_2g+1 is the set of all monic irreducible polynomials of degree 2g+1 in 𝔽_q[t]. In particular, when κ=0, this gives the corresponding result at the central point σ=1/2. Our result extends the recent work of Darbar and Maiti (2024) and yields an improved lower bound for the extreme values in this family. In addition, for 1/2<σ<1 and sufficiently large n, we study the extreme values of L(σ,χ_P), where P∈𝒫_n, and obtain an improved lower bound compared with the result of Lumley (2021).
ER  -