Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2019; 34(4): 1105-1115

Online first article August 27, 2019      Printed October 31, 2019

https://doi.org/10.4134/CKMS.c180414

Copyright © The Korean Mathematical Society.

Generalizations of number-theoretic sums

Narakorn Rompurk Kanasri, Patchara Pornsurat, Yanapat Tongron

Khon Kaen University; Khon Kaen University; Khon Kaen University

Abstract

For positive integers $n$ and $k$, let $S_k(n)$ and $S'_k(n)$ be the sums of the elements in the finite sets $\left\{x^k : 1\leq x\leq n, (x, n)=1\right\}$ and $\left\{x^k : 1\leq x\leq n/2, (x, n)=1\right\}$, respectively. The formulae for both $S_k(n)$ and $S'_k(n)$ are established. The explicit formulae when $k=1, 2, 3$ are also given.

Keywords: arithmetic function, Euler's phi-function, M\"obius function, M\"obius inversion formula, sum of power of integers

MSC numbers: 11A25

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