Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-2010-10560-9
Published electronically: August 30, 2010
MathSciNet review: 2748421
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • 1. J. Bourgain, Exponential sum estimates over subgroups of $ {\mathbb{Z}}_q^*$, $ q$ arbitrary, J. Analyse Math., 97 (2005), 317-355. MR 2274981 (2007j:11103)
  • 2. K. Girstmair, A ``popular'' class number formula, American Math. Monthly, 101 (1994), no. 10, 997-1001. MR 1304325 (95k:11003)
  • 3. K. Girstmair, The digits of $ 1/p$ in connection with class number factors, Acta Arith., 67 (1994), no. 4, 381-386. MR 1301825 (96g:11134)
  • 4. K. Girstmair, Periodische Dezimalbrüche - was nicht jeder darüber weiss, Jahrbruch Überblicke Mathematik, 1995, 163-179, Vieweg, Braunschweig, 1995. MR 1342348 (96f:11012)
  • 5. A. Ivić, Two inequalities for the sum of divisors functions, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak., 7 (1977), 17-22. MR 0505991 (58:21911)
  • 6. M. Ram Murty, Problems in Analytic Number Theory, Graduate Texts in Mathematics, Springer, New York, 2008, 2nd edition. MR 2376618 (2008j:11001)
  • 7. M. Ram Murty and Kaneenika Sinha, Effective equidistribution of eigenvalues of Hecke operators, Journal of Number Theory, 129 (2009), 681-714. MR 2488597
  • 8. L. Washington, Introduction to cyclotomic fields, second edition, Graduate Texts in Mathematics, 83, Springer-Verlag, New York, 1997. MR 1421575 (97h:11130)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11A07, 11R29

Retrieve articles in all journals with MSC (2010): 11A07, 11R29


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: https://doi.org/10.1090/S0002-9939-2010-10560-9
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.

American Mathematical Society