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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Borel's conjecture and the transcendence of the Iwasawa power series

Author(s): Hae-Sang Sun
Journal: Proc. Amer. Math. Soc. 138 (2010), 1955-1963.
MSC (2010): Primary 11K16, 11R23, 11R42
Posted: February 4, 2010
MathSciNet review: 2596029
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Abstract | References | Similar articles | Additional information

Abstract: We deduce the transcendence of the Iwasawa power series from Borel's conjecture, namely, the normality of the irrational algebraic $ p$-adic integers.

References:

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J.-P. Allouche and J. Shallit, Automatic sequences, Cambridge Univ. Press, Cambridge, 2003. MR 1997038 (2004k:11028)

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B. Anglès, On some $ p$-adic power series attached to the arithmetic of $ \mathbb{Q}(\zeta_p)$, J. Number Theory 122 (2007), no. 1, 221-246. MR 2287121 (2008g:11182)

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B. Anglès, On the $ p$-adic Leopoldt transform of a power series, Acta Arith. 134 (2008), no. 4, 349-367. MR 2449158

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É. Borel, Sur les chiffres décimaux de $ \sqrt{2}$ et divers problèmes de probabilités en chaîne. C. R. Acad. Sci. Paris 230 (1950), 591-593. MR 0034544 (11:605d)

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H. Hida, Elementary Theory of $ L$-Functions and Eisenstein Series, London Math. Soc. Student Texts, 26, Cambridge Univ. Press, Cambridge, UK, 1993. MR 1216135 (94j:11044)

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K. Iwasawa, Lectures on $ p$-adic $ L$-functions, Ann. of Math. Studies, 74. Princeton Univ. Press, 1972. MR 0360526 (50:12974)

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L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Pure and Applied Mathematics, John Wiley $ \&$ Sons, New York-London-Sydney, 1974. MR 0419394 (54:7415)

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H.-S. Sun, Derivative of power series attached to $ \Gamma$-transform of $ p$-adic measures, J. Number Theory (2009), in press, doi:10.1016/j.jnt.2009.07.001

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Additional Information:

Hae-Sang Sun
Affiliation: Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun-gu, Seoul 130-722, Republic of Korea
Email: haesang@kias.re.kr

DOI: 10.1090/S0002-9939-10-10287-1
PII: S 0002-9939(10)10287-1
Received by editor(s): August 17, 2009,
Received by editor(s) in revised form: September 15, 2009, and October 5, 2009
Posted: February 4, 2010
Additional Notes: The manuscript was prepared while the author was visiting Department of the Mathematics, Université de Caen. He thanks the department for their support and hospitality. He also thanks the referee for the valuable suggestions and corrections.
Communicated by: Ken Ono
Copyright of article: Copyright 2010, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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