ON A SUM INVOLVING CERTAIN ARITHMETIC FUNCTIONS ON PIATETSKI–SHAPIRO AND BEATTY SEQUENCES

Abstract

Let 𝑐, 𝛼, 𝛽 ∈ R be such that 1 1 is irrational and with bounded partial quotients, 𝛽 ∈ [0, 𝛼). In this paper, we study asymptotic behaviour of the summations of the form Σ_(n≤N) (f([n^c]))/([n^c]) and Σ_(n≤N) (f([𝛼n+𝛽]))/([𝛼n+𝛽]) , where f is the Euler totient function 𝜑, Dedekind function Ψ, sum-of-divisors function 𝜎, or the alternating sum-of-divisors function 𝜎_(𝑎𝑙𝑡).

Keywords

• arithmetic function
• beatty sequence
• piatetski–shapiro sequence