Abel Summation of Ramanujan-Fourier Series

TL;DR

This paper demonstrates that Abel summation can be used to establish a weak Wiener-Khinchin type formula for arithmetic functions with Ramanujan-Fourier expansions, linking convolutions to their Fourier coefficients.

Contribution

It introduces a novel application of Abel summation to Ramanujan-Fourier series, proving a weak form of the Wiener-Khinchin formula for these functions.

Findings

Convolution of arithmetic functions with Ramanujan-Fourier expansions are Abel-summable.

The Abel sum involves only the Ramanujan-Fourier coefficients.

The results extend the understanding of Ramanujan-Fourier series and their summability properties.

Abstract

Using Abel summation the paper proves a weak form of the Wiener-Khinchin formula for arithmetic functions with point-wise convergent Ramanujan-Fourier expansions. The main result is that the convolution of most arithmetic functions possessing an R-F expansion are Abel-summable to a result involving only the Ramanujan-Fourier coefficients of the R-F expansion(s).