REMARKS ON THE ARITHMETICAL FUNCTION $a_{p}(n)$

Abstract

In this paper,for any arbitrary  two prime numbers $p$ and $q$ the relationship between the corresponding  arithmetic functions  $(a_p(n))$ and  $(a_q(n))$ are investigated.~Furthermore,  the  general formula for  statistical density of all sets on which the two arithmetic functions have the same value  also established.

Keywords

• statistical density,Arithmetical functions

References

1. \bibitem{1} Fast, H. , (1951) Sur la convergence statistique., \textit{Colloq.Math}, VOL, 2, 241--244.
2. \bibitem{2} Schoenberg, I. J., (1959) The integrability of certain functions and related summability methods, matrix characterization of statistical convergence, \textit{Amer.Math.}, VOL, 66, 361--375.
3. \bibitem{3} Zygmund, A. (1979) \textit{Trigonometric series}, 2nd., Ed. Vol. II, Cambridge Univ. press, London and New York.
4. \bibitem{4} {\v{S}}al{\'a}t, T., (1994) On the function $ a_{p},~p^{a_{p}(n)} \backslash\backslash ~ n ~(n>1)$ , \textit{Mathematica Slovaca}, VOL, 44, Num, 2 143--151.
5. \bibitem{5} Feh{\'e}ra, Zolt{\'a}n and L{\'a}szl{\'o}a, B{\'e}la and Ma{\v{c}}ajb, Martin and {\v{S}}al{\'a}tb, Tibor, 2006,Remarks on arithmetical functions $ap (n)$, $\gamma$ (n), $\tau$ (n),Annales Mathematicae et Informaticae,Vol 33,35-43
6. \bibitem{6} Abdu Awel and M. K\"{u}\c{c}\"{u}kaslan ,A Note On Limit And Cluster Points Of The Arithmetical Functions,J.Indones.Math.Soc(under review)
7. \bibitem{10} Milan, P. (2017) \textit{DENSITY AND RELATED TOPICS}, Mathematics Institute slovak Acadamic of Sciences