Generalizations of Number-Theoretic Sums
Commun. Korean Math. Soc.
Published online June 3, 2019
Narakorn Rompurk Kanasri, Patchara Pornsurat, and Yanapat Tongron
Department of Mathematics, Faculty of Science, Khon Kaen University, Thailand.
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, 11L99
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