Geodesic
On a certain class of arithmetic functions
Mathematica Bohemica, Tome 142 (2017) no. 1, pp. 21-25
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

A homothetic arithmetic function of ratio $K$ is a function $f\colon \mathbb {N}\rightarrow R$ 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 $f$.
A homothetic arithmetic function of ratio $K$ is a function $f\colon \mathbb {N}\rightarrow R$ 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 $f$.
DOI : 10.21136/MB.2017.0071-14
Classification : 11A25, 11B99, 11N37
Keywords: arithmetic function; periodic function; homothetic function 
@article{10_21136_MB_2017_0071_14,
     author = {Oller-Marc\'en, Antonio M.},
     title = {On a certain class of arithmetic functions},
     journal = {Mathematica Bohemica},
     pages = {21--25},
     year = {2017},
     volume = {142},
     number = {1},
     doi = {10.21136/MB.2017.0071-14},
     mrnumber = {3619984},
     zbl = {06738567},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2017.0071-14/}
}
TY  - JOUR
AU  - Oller-Marcén, Antonio M.
TI  - On a certain class of arithmetic functions
JO  - Mathematica Bohemica
PY  - 2017
SP  - 21
EP  - 25
VL  - 142
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2017.0071-14/
DO  - 10.21136/MB.2017.0071-14
LA  - en
ID  - 10_21136_MB_2017_0071_14
ER  - 
%0 Journal Article
%A Oller-Marcén, Antonio M.
%T On a certain class of arithmetic functions
%J Mathematica Bohemica
%D 2017
%P 21-25
%V 142
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2017.0071-14/
%R 10.21136/MB.2017.0071-14
%G en
%F 10_21136_MB_2017_0071_14
Oller-Marcén, Antonio M. On a certain class of arithmetic functions. Mathematica Bohemica, Tome 142 (2017) no. 1, pp. 21-25. doi: 10.21136/MB.2017.0071-14

[1] Apostol, T. M.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics Springer, New York (1976). | MR | JFM

[2] Cohen, E.: A property of Dedekind's {$\psi $}-function. Proc. Am. Math. Soc. 12 (1961), 996. | DOI | MR | JFM

[3] Ghiyasi, A. K.: 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 | MR | JFM

[4] Grau, J. M., Oller-Marcén, A. M.: 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 | MR | JFM

[5] Pilehrood, T. Hessami, Pilehrood, K. Hessami: On a conjecture of Erdős. Math. Notes 83 (2008), 281-284 translation from Mat. Zametki83 2008 312-315. | DOI | MR | JFM

[6] Ji, Q.-Z., Ji, C.-G.: On the periodicity of some Farhi arithmetical functions. Proc. Am. Math. Soc. 138 (2010), 3025-3035. | DOI | MR | JFM

[7] Rausch, U.: Character sums in algebraic number fields. J. Number Theory 46 (1994), 179-195. | DOI | MR | JFM

[8] Steuding, J.: 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 | JFM

Cité par Sources :