The linear independence of $1$, $\zeta(2)$, and $L(2,\chi_{-3})$

Frank Calegari; Vesselin Dimitrov; Yunqing Tang

The linear independence of $1$, $\zeta(2)$, and $L(2,\chi_{-3})$

Number Theory 2024-09-18 v2

Authors: Frank Calegari , Vesselin Dimitrov , Yunqing Tang

Abstract

We prove the irrationality of the classical Dirichlet L-value $L(2,\chi_{-3})$ . The argument applies a new kind of arithmetic holonomy bound to a well-known construction of Zagier. In fact our work also establishes the $\mathbf{Q}$ -linear independence of $1$ , $\zeta(2)$ , and $L(2,\chi_{-3})$ . We also give a number of other applications of our method to other problems in irrationality.

Keywords

Dirichlet L-value irrationality linear independence arithmetic holonomy bound Zagier construction

Cite

@article{arxiv.2408.15403,
  title  = {The linear independence of $1$, $\zeta(2)$, and $L(2,\chi_{-3})$},
  author = {Frank Calegari and Vesselin Dimitrov and Yunqing Tang},
  journal= {arXiv preprint arXiv:2408.15403},
  year   = {2024}
}

Comments

218 pages, comments welcome, minor expository changes, this is the submitted version of the paper

The linear independence of $1$, $\zeta(2)$, and $L(2,\chi_{-3})$

Abstract

Keywords

Cite

Comments

Related papers