On Summatory Totient Functions

TL;DR

This paper establishes bounds for the leading term of summatory totient functions involving powers of integers and the totient function, across different parameter ranges.

Contribution

It provides new bounds for the summatory totient function with powers, extending understanding in number theory.

Findings

Derived bounds for summatory totient functions in various parameter ranges

Extended previous results to more general power cases

Enhanced understanding of the asymptotic behavior of totient sums

Abstract

The lower and upper bounds are found for the leading term of summatory totient function $\sum_{k\leq N}k^u\phi^v(k)$ in various ranges of $u\in{\mathbb R}$ and $v\in{\mathbb Z}$.