Hypothetical connection of the theta functions of Dirichlet characters with the real cyclotomic fields
Abstract
We consider a possible approach to the Lindelöf hypothesis for Dirichlet L-functions. It is based on a special form of the functional equation for the corresponding theta functions. To estimate Lχ(0.5+it) we need to solve certain systems of linear equations. The entries to the corresponding matrices are formed by the summands to the series for theta functions and their derivatives. Numerical data suggest that the inverse matrices have a deep structure and allow us to state a number of conjectures. In particular, it seems that for a character modulo q certain entries to the inverse matrices tend to finite limits when the sizes of the matrices run over arithmetical progressions with step 2q. Moreover, these limits belong to the real cyclotomic field ℚ(cos(π/q)) (up to a scaling factor of √q).
Related
- Irrationality of values of L-functions for real Dirichlet characters
- Angular Monotonicity of Completed Dirichlet L-Functions via the Three-Term Zeta Decomposition: A Proof of the Generalized Riemann Hypothesis for Non-Principal Characters
- Upper Bounds on the de Bruijn-Newman constant for Dirichlet L-functions with primitive real-valued characters
- String functions and cyclotomic Hecke algebras
- Distribution of central values of Hecke L-functions of n-th order characters
BibTeX
@misc{matiyasevich2026hypothetical,
title = {{Hypothetical connection of the theta functions of Dirichlet characters with the real cyclotomic fields}},
author = {Yuri Matiyasevich},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17478},
eprint = {2609.17478},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17478},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Hypothetical connection of the theta functions of Dirichlet characters with the real cyclotomic fields
AU - Yuri Matiyasevich
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.17478
UR - https://arxiv.org/abs/2609.17478
AB - We consider a possible approach to the Lindelöf hypothesis for Dirichlet L-functions. It is based on a special form of the functional equation for the corresponding theta functions. To estimate L_χ(0.5+it) we need to solve certain systems of linear equations. The entries to the corresponding matrices are formed by the summands to the series for theta functions and their derivatives. Numerical data suggest that the inverse matrices have a deep structure and allow us to state a number of conjectures. In particular, it seems that for a character modulo q certain entries to the inverse matrices tend to finite limits when the sizes of the matrices run over arithmetical progressions with step 2q. Moreover, these limits belong to the real cyclotomic field ℚ(cos(π/q)) (up to a scaling factor of √q).
ER -