×

zbMATH — the first resource for mathematics

Arithmetic subderivatives: discontinuity and continuity. (English) Zbl 1448.11008
Summary: We first prove that any arithmetic subderivative of a rational number defines a function that is everywhere discontinuous in a very strong sense. Second, we show that although the restriction of this function to the set of integers is continuous (in the relative topology), it is not Lipschitz continuous. Third, we see that its restriction to a suitable infinite set is Lipschitz continuous. This follows from the solutions of certain arithmetic differential equations.

MSC:
11A25 Arithmetic functions; related numbers; inversion formulas
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
Software:
OEIS
PDF BibTeX XML Cite
Full Text: Link
References:
[1] T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976.
[2] E. J. Barbeau, Remarks on an arithmetic derivative, Canad. Math. Bull. 4 (1961), 117- 122. · Zbl 0101.03702
[3] P. Haukkanen, J. K. Merikoski, and T. Tossavainen, On arithmetic partial differential equations, J. Integer Sequences 19 (2016),Article 16.8.6.
[4] J. Koviˇc, The arithmetic derivative and antiderivative, J. Integer Sequences 15 (2012), Article 12.3.8.
[5] J. K. Merikoski, P. Haukkanen, and T. Tossavainen, Arithmetic subderivatives and Leibniz-additive functions, Ann. Math. Informat. 50 (2019). Available athttp://ami. ektf.hu.
[6] J. Mingot Shelly, Una cuesti´on de la teor´ıa de los n´umeros, Asociaci´on Espa˜nola, Granada (1911), 1-12.
[7] R. K. Pandey and R. Saxena, On some conjectures about arithmetic partial differential equations, J. Integer Sequences 20 (2017),Article 17.5.2.
[8] V. Ufnarovski and B. ˚Ahlander, How to differentiate a number, J. Integer Sequences 6 (2003),Article 03.3.4.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.