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Arithmetic subderivatives: \(p\)-adic discontinuity and continuity. (English) Zbl 1453.11004
Summary: In a previous paper, we proved that the arithmetic subderivative \(D_S\) is discontinuous at any rational point with respect to the ordinary absolute value. In the present paper, we study this question with respect to the \(p\)-adic absolute value. In particular, we show that \(D_S\) is in this sense continuous at the origin if \(S\) is finite or \(p \not\in S\).
MSC:
11A25 Arithmetic functions; related numbers; inversion formulas
11S82 Non-Archimedean dynamical systems
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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References:
[1] T. M. Apostol,Introduction to Analytic Number Theory, Springer, 1976. · Zbl 0335.10001
[2] A. Baker, An introduction top-adic numbers andp-adic analysis, lecture notes, University of Glasgow, 2020,http://www.maths.gla.ac.uk/ ajb/course-notes.html.
[3] E. J. Barbeau, Remarks on an arithmetic derivative,Canad. Math. Bull.4(1961), 117-122. · Zbl 0101.03702
[4] J. Fan and S. Utev, The Lie bracket and the arithmetic derivative,J. Integer Sequences23(2020),Article 20.2.5. · Zbl 1444.11012
[5] R. L. Graham, D. E. Knuth, and O. Patashnik,Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 2nd ed., 1994. · Zbl 0836.00001
[6] P. Haukkanen, M. Mattila. J. K. Merikoski, and T. Tossavainen, Perpendicularity in an Abelian group,Internat. J. Math. Math. Sci.2013, Article 983607. · Zbl 1264.51012
[7] P. Haukkanen, J. K. Merikoski, and T. Tossavainen, On arithmetic partial differential equations,J. Integer Sequences19(2016),Article 16.8.6. · Zbl 1353.11007
[8] P. Haukkanen, J. K. Merikoski, and T. Tossavainen, Arithmetic subderivatives: Discontinuity and continuity,J. Integer Sequences22(2019),Article 19.7.4. · Zbl 1448.11008
[9] P. Haukkanen, J. K. Merikoski, and T. Tossavainen, Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative,Math. Comm.25(2020), 107-115,https:// www.mathos.unios.hr/mc/index.php/mc/article/view/3206. · Zbl 07200858
[10] J. Koviˇc, The arithmetic derivative and antiderivative,J. Integer Sequences15(2012), Article 12.3.8.
[11] J. K. Merikoski, P. Haukkanen, and T. Tossavainen, Arithmetic subderivatives and Leibniz-additive functions,Ann. Math. Informat.50(2019), 145-157,http://ami. ektf.hu. · Zbl 1444.11015
[12] J. Mingot Shelly, Una cuesti´on de la teor´ıa de los n´umeros,Asociaci´on Espa˜nola, Granada(1911), 1-12.
[13] V.
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