Czechoslovak Mathematical Journal, Vol. 71, No. 1, pp. 283-308, 2021


On $p$-adic Euler constants

Abhishek Bharadwaj

Received July 29, 2019.   Published online October 9, 2020.

Abstract:  The goal of this article is to associate a $p$-adic analytic function to the Euler constants $\gamma_p (a, F)$, study the properties of these functions in the neighborhood of $s=1$ and introduce a $p$-adic analogue of the infinite sum $\sum_{n \ge1} f(n) / n$ for an algebraic valued, periodic function $f$. After this, we prove the theorem of Baker, Birch and Wirsing in this setup and discuss irrationality results associated to $p$-adic Euler constants generalising the earlier known results in this direction. Finally, we define and prove certain properties of $p$-adic Euler-Briggs constants analogous to the ones proved by Gun, Saha and Sinha.
Keywords:  $p$-adic Euler-Lehmer constant; linear forms in logarithms
Classification MSC:  11J91


References:
[1] T. M. Apostol: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics. Springer, New York (1976). DOI 10.1007/978-1-4757-5579-4 | MR 0434929 | Zbl 0335.10001
[2] A. Baker, B. J. Birch, E. A. Wirsing: On a problem of Chowla. J. Number Theory 5 (1973), 224-236. DOI 10.1016/0022-314X(73)90048-6 | MR 0340203 | Zbl 0267.10065
[3] A. T. Bharadwaj: A short note on generalized Euler-Briggs constants. Int. J. Number Theory 16 (2020), 823-839. DOI 10.1142/S1793042120500426 | MR 4093385 | Zbl 07205429
[4] A. Brumer: On the units of algebraic number fields. Mathematika, Lond. 14 (1967), 121-124. DOI 10.1112/S0025579300003703 | MR 0220694 | Zbl 0171.01105
[5] T. Chatterjee, S. Gun: The digamma function, Euler-Lehmer constants and their $p$-adic counterparts. Acta Arith. 162 (2014), 197-208. DOI 10.4064/aa162-2-4 | MR 3167891 | Zbl 1285.11105
[6] H. Cohen: Number Theory. Volume II. Analytic and Modern Tools. Graduate Texts in Mathematics 240. Springer, New York (2007). DOI 10.1007/978-0-387-49894-2 | MR 2312338 | Zbl 1119.11002
[7] J. Diamond: The $p$-adic log gamma function and $p$-adic Euler constants. Trans. Am. Math. Soc. 233 (1977), 321-337. DOI 10.1090/S0002-9947-1977-0498503-9 | MR 0498503 | Zbl 0382.12008
[8] S. Gun, V. K. Murty, E. Saha: Linear and algebraic independence of generalized Euler-Briggs constants. J. Number Theory 166 (2016), 117-136. DOI 10.1016/j.jnt.2016.02.004 | MR 3486268 | Zbl 1415.11100
[9] S. Gun, E. Saha, S. B. Sinha: Transcendence of generalized Euler-Lehmer constants. J. Number Theory 145 (2014), 329-339. DOI 10.1016/j.jnt.2014.06.010 | MR 3253307 | Zbl 1325.11071
[10] N. Koblitz: Interpretation of the $p$-adic log gamma function and Euler constants using the Bernoulli measure. Trans. Am. Math. Soc. 242 (1978), 261-269. DOI 10.1090/S0002-9947-1978-0491622-3 | MR 0491622 | Zbl 0358.12010
[11] T. Kubota, H. W. Leopoldt: Eine $p$-adische Theorie der Zetawerte. I. Einführung der $p$-adischen Dirichletschen $L$-Funktionen. J. Reine Angew. Math. 214/215 (1964), 328-339. (In German.) DOI 10.1515/crll.1964.214-215.328 | MR 0163900 | Zbl 0186.09103
[12] S. Lang: Algebraic Number Theory. Graduate Texts in Mathematics 110. Springer, New York (1994). DOI 10.1007/978-1-4612-0853-2 | MR 1282723 | Zbl 0811.11001
[13] D. H. Lehmer: Euler constants for arithmetical progressions. Acta Arith. 27 (1975), 125-142. DOI 10.4064/aa-27-1-125-142 | MR 0369233 | Zbl 0302.12003
[14] Y. Morita: A $p$-adic analogue of the $\Gamma$-function. J. Fac. Sci., Univ. Tokyo, Sect. I A 22 (1975), 255-266. MR 0424762 | Zbl 0308.12003
[15] Y. Morita: On the Hurwitz-Lerch $L$-functions. J. Fac. Sci., Univ. Tokyo, Sect. I A 24 (1977), 29-43. MR 0441924 | Zbl 0356.12019
[16] M. R. Murty, N. Saradha: Transcendental values of the digamma function. J. Number Theory 125 (2007), 298-318. DOI 10.1016/j.jnt.2006.09.017 | MR 2332591 | Zbl 1222.11097
[17] M. R. Murty, S. Pathak: Special values of derivatives of $L$-series and generalized Stieltjes constants. Acta Arith. 184 (2018), 127-138. DOI 10.4064/aa170615-13-3 | MR 3841150 | Zbl 1421.11057
[18] M. R. Murty, N. Saradha: Transcendental values of the $p$-adic digamma function. Acta Arith. 133 (2008), 349-362. DOI 10.4064/aa133-4-4 | MR 2457265 | Zbl 1253.11077
[19] T. Okada: Dirichlet series with periodic algebraic coefficients. J. Lond. Math. Soc., II. Ser. 33 (1986), 13-21. DOI 10.1112/jlms/s2-33.1.13 | MR 0829383 | Zbl 0589.10034
[20] A. M. Robert: A Course in $p$-Adic Analysis. Graduate Texts in Mathematics 198. Springer, New York (2000). DOI 10.1007/978-1-4757-3254-2 | MR 1760253 | Zbl 0947.11035
[21] L. C. Washington: Introduction to Cyclotomic Fields. Graduate Texts in Mathematics 83. Springer, New York (1982). DOI 10.1007/978-1-4684-0133-2 | MR 0718674 | Zbl 0484.12001

Affiliations:   Abhishek Bharadwaj, Chennai Mathematical Institute, H1 State Industries Promotion Corporation of Tamil Nadu Limited IT Park, Old Mahabalipuram Road, Siruseri, Kellambakkam, 603103 Chennai, India, e-mail: abhitvt@cmi.ac.in


 
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