Mathematica Bohemica, Vol. 142, No. 1, pp. 21-25, 2017
On a certain class of arithmetic functions
Antonio M. Oller-Marcén
Received October 20, 2014. First published October 18, 2016.
Abstract: A homothetic arithmetic function of ratio $K$ is a function $f \mathbb{N}\rightarrow R$ such that $f(Kn)=f(n)$ for every $n\in\mathbb{N}$. Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of $f(\mathbb{N})$ in terms of the period and the ratio of $f$.
Keywords: arithmetic function; periodic function; homothetic function
References: [1] T. M. Apostol: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics Springer, New York (1976). MR 0434929 | Zbl 0335.10001 [2] E. Cohen: A property of Dedekind's {$\psi $}-function. Proc. Am. Math. Soc. 12 (1961), 996. DOI 10.2307/2034413 | MR 0132717 | Zbl 0144.28201 [3] A. K. Ghiyasi: Constants in inequalities for the mean values of some periodic arithmetic functions. Mosc. Univ. Math. Bull. 63 (2008), 265-269 translation from Vest. Mosk. Univ. Mat. Mekh. 63 (2008), 44-48. DOI 10.3103/S0027132208060077 | MR 2517021 | Zbl 1304.11085 [4] J. M. Grau, A. M. Oller-Marcén: On the last digit and the last non-zero digit of $n^n$ in base {$b$}. Bull. Korean Math. Soc. 51 (2014), 1325-1337. DOI 10.4134/BKMS.2014.51.5.1325 | MR 3267232 | Zbl 1302.11002 [5] T. Hessami Pilehrood, K. Hessami Pilehrood: On a conjecture of Erdős. Math. Notes 83 (2008), 281-284 translation from Mat. Zametki83 (2008), 312-315. DOI 10.1134/S0001434608010306 | MR 2431590 | Zbl 1157.11031 [6] Q.-Z. Ji, C.-G. Ji: On the periodicity of some Farhi arithmetical functions. Proc. Am. Math. Soc. 138 (2010), 3025-3035. DOI 10.1090/S0002-9939-10-10408-0 | MR 2653927 | Zbl 1246.11013 [7] U. Rausch: Character sums in algebraic number fields. J. Number Theory 46 (1994), 179-195. DOI 10.1006/jnth.1994.1011 | MR 1269251 | Zbl 0795.11033 [8] J. Steuding: Dirichlet series associated to periodic arithmetic functions and the zeros of Dirichlet {$L$}-functions. Analytic and Probabilistic Methods in Number Theory. Proc. Int. Conf., Palanga, Lithuania, 2001 A. Dubickas et al. TEV, Vilnius (2002), 282-296. MR 1964871 | Zbl 1035.11041
Affiliations: Antonio Miguel Oller-Marcén, Centro Universitario de la Defensa de Zaragoza, Academia General Militar. Ctra. de Huesca s/n., C. P. 50090 Zaragoza, Spain, e-mail: oller@unizar.es
