A simple proof of generalizations of number-theoretic sums

Abstract

For positive integers 𝑘, 𝑚, and 𝑛, let 𝑆𝑘 𝑚(𝑛) be the sum of all elements in the finite set {𝑥 𝑘 : 1 ≤ 𝑥 ≤ 𝑛⁄𝑚 , (𝑥, 𝑛) = 1}. The formula for 𝑆𝑘 𝑚(𝑛) is established and simpler formulae for 𝑆𝑘 𝑚(𝑛) under some conditions on 𝑚 and 𝑛 are verified. The explicit formulae for 𝑆1 2 𝑎 (𝑛) and 𝑆2 2 𝑎 (𝑛), where 2 𝑎 |𝑛 and 𝑎 ≥ 1, are also provided.

Keywords

• arithmetic function
• euler’s phi-function
• möbius function
• möbius inversion formula