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A certain character twisted average value of the digits of rational numbers and the class numbers of imaginary quadratic fields

Volume 208 / 2023

Yoshinori Mizuno Acta Arithmetica 208 (2023), 215-233 MSC: Primary 11A63; Secondary 11R29. DOI: 10.4064/aa220114-28-5 Published online: 21 August 2023


We give a closed formula for a certain character twisted average value of the digits of the base $g$ expansion of $a/m$ ($2\leq m\in \mathbb N$, $a \in \mathbb N$, $1\leq a \lt m$, $(a,m)=1$). The closed formula involves the first generalized Bernoulli numbers, and we can apply it to the class number of any imaginary quadratic field. In particular, Girstmair’s formula about the class number of $\mathbb Q(\sqrt{-p})$ with a prime $p\equiv 3$ (mod 4) is generalized to the case $p\equiv 1$ (mod 4).


  • Yoshinori MizunoGraduate School of Technology, Industrial and Social Sciences
    Tokushima University
    Tokushima, 770-8506, Japan

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