The first moment of quadratic Dirichlet L-functions in the even hyperelliptic ensemble

@misc{jung2026first,
  title = {{The first moment of quadratic Dirichlet $L$-functions in the even hyperelliptic ensemble}},
  author = {Hwanyup Jung},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12764},
  eprint = {2609.12764},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12764},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - The first moment of quadratic Dirichlet L-functions in the even hyperelliptic ensemble
AU  - Hwanyup Jung
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12764
UR  - https://arxiv.org/abs/2609.12764
AB  - We establish an asymptotic formula for the first moment of the central values of quadratic Dirichlet L-functions in the even hyperelliptic ensemble ℋ_2g+2 over 𝔽_q[x], for every fixed odd prime power q. Working with the completed L-function yields an exact two-term central-value formula with truncation levels g and g−1; combining the two truncations before the square-dual contour shifts leaves the three cubic points z^3 = q^−4 as the only secondary poles, and they contribute the secondary term q^2g/3+2(a_g g + b_g) with an error O_ε(q^g(1+ε)/2). The coefficients are real and m ↦ (a_m, b_m) has minimal period exactly three, so the coefficient of g at the order q^2g/3 depends on g modulo 3, in contrast with the earlier even-degree formulas; the difference is confirmed numerically.
ER  -