The β=2 partition function of Dirichlet L-functions and typical maxima
Abstract
We study the β=2 partition function ∫|h| ≤ log θ q/2|L(1/2+ih,χ)|2dh for typical Dirichlet characters modulo a large prime q and θ∈ (−1/3,0] motivated by a q-analogue of the Saksman--Webb conjectures. When θ<0, we use an extension of Harper's randomisation argument with different localisations to recover moment upper bounds consistent with the critical normalisation predicted there. As an application, we prove that for $q(1-o(1))$ Dirichlet characters modulo q with U(q) → ∞, max|h| ≤ 1/2|L(1/2+ih,χ)|≪ (eU(q)log qlog3 q(log4 q)1/2)/((log2 q)3/4), establishing an upper bound matching the predictions of the q-analogue of the Fyodorov--Hiary--Keating conjectures up to second order. Our proofs adapt Harper's argument in \cite{harper2019partition}.
Related
- Isolated zeros of the Riemann zeta function and zeros of linear combinations of Dirichlet L-functions
- Asymptotics for partition functions of random normal matrix models with singularities
- The Calkin--Wilf problems on ℓ-ary partition functions
- Efficient Computation and Congruences for Colored Partition Functions
- The First Moment of Dirichlet L-functions over Generators of the Character Group
BibTeX
@misc{atherfold2026partition,
title = {{The $β=2$ partition function of Dirichlet $L$-functions and typical maxima}},
author = {Christopher Atherfold},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.09906},
eprint = {2608.09906},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09906},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The β=2 partition function of Dirichlet L-functions and typical maxima
AU - Christopher Atherfold
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2608.09906
UR - https://arxiv.org/abs/2608.09906
AB - We study the β=2 partition function ∫_|h| ≤ log ^θ q/2|L(1/2+ih,χ)|^2dh for typical Dirichlet characters modulo a large prime q and θ∈ (−1/3,0] motivated by a q-analogue of the Saksman--Webb conjectures. When θ<0, we use an extension of Harper's randomisation argument with different localisations to recover moment upper bounds consistent with the critical normalisation predicted there. As an application, we prove that for $q(1-o(1))$ Dirichlet characters modulo q with U(q) → ∞, max_|h| ≤ 1/2|L(1/2+ih,χ)|≪ (e^U(q)log qlog_3 q(log_4 q)^1/2)/((log_2 q)^3/4), establishing an upper bound matching the predictions of the q-analogue of the Fyodorov--Hiary--Keating conjectures up to second order. Our proofs adapt Harper's argument in \cite{harper2019partition}.
ER -