Addendum to "Arithmetic exponent pairs of algebraic trace functions and applications"

TL;DR

This paper provides a detailed proof of a key inequality from a previous work on arithmetic exponent pairs for algebraic trace functions, clarifying complex averaging techniques involved.

Contribution

It offers an in-depth proof of an important inequality from the authors' earlier research, enhancing understanding of their methods.

Findings

Detailed proof of inequality (9.14) provided

Clarification of averaging techniques in arithmetic functions

Supports the application of arithmetic exponent pairs

Abstract

This addendum devotes to a detailed proof for the inequality (9.14) in our joint work: Arithmetic exponent pairs for algebraic trace functions and applications, with an appendix by Will Sawin, arXiv:1603.07060 [math.NT], which will appear in Algebra and Number Theory. We do not intend to publish this addendum in any journals; arXiv should be a good place for those reader who want to find such details. The proof involves various averages of arithmetic functions.