Harmonic Analysis on the Positive Rationals I: Basic Results

TL;DR

This paper investigates the mean values of multiplicative functions on arithmetic progressions with large differences, with applications to primes in progressions and automorphic forms.

Contribution

It provides initial results on harmonic analysis of positive rationals, focusing on estimating mean values of multiplicative functions in complex settings.

Findings

Estimates for mean values of multiplicative functions on large arithmetic progressions.

Applications to understanding primes in arithmetic progressions.

Insights into Fourier coefficients of automorphic cusp forms.

Abstract

This is the first of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression and to the Fourier coefficients of automorphic cusp forms.