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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The class number of $ \mathbb{Q}(\sqrt{-p})$ and digits of $ 1/p$


Authors: M. Ram Murty and R. Thangadurai
Journal: Proc. Amer. Math. Soc. 139 (2011), 1277-1289
MSC (2010): Primary 11A07, 11R29
Published electronically: August 30, 2010
MathSciNet review: 2748421
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Abstract: Let $ p$ be a prime number such that $ p\equiv 1\pmod{r}$ for some integer $ r >1$. Let $ g>1$ be an integer such that $ g$ has order $ r$ in $ \left(\mathbb{Z}/p\mathbb{Z}\right)^*$. Let

$\displaystyle \frac 1p = \sum_{k=1}^\infty\frac{x_k}{g^k}$

be the $ g$-adic expansion. Our result implies that the ``average'' digit in the $ g$-adic expansion of $ 1/p$ is $ (g-1)/2$ with error term involving the generalized Bernoulli numbers $ B_{1,\chi}$ (where $ \chi$ is a character modulo $ p$ of order $ r$ with $ \chi(-1) = -1)$. Also, we study, using Bernoulli polynomials and Dirichlet $ L$-functions, the set equidistribution modulo $ 1$ of the elements of the subgroup $ H_n$ of $ \left(\mathbb{Z}/{n\mathbb{Z}}\right)^*$ as $ n\to\infty$ whenever $ \vert H_n\vert/\sqrt{n} \to \infty$.


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Additional Information

M. Ram Murty
Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada
Email: murty@mast.queensu.ca

R. Thangadurai
Affiliation: Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahbad, 211019, India
Email: thanga@hri.res.in

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10560-9
PII: S 0002-9939(2010)10560-9
Keywords: Digits, Bernoulli polynomials, Dirichlet $L$-functions, equidistribution modulo 1
Received by editor(s): January 18, 2010
Received by editor(s) in revised form: April 26, 2010
Published electronically: August 30, 2010
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.