Digit sums and generating functions

TL;DR

This paper explores the mathematical properties of digit sums in various bases, linking them to q-series, Lambert series, and Dirichlet series, revealing new connections in number theory and generating functions.

Contribution

It introduces a novel connection between digit sum operations and advanced generating functions like Lambert and Dirichlet series, expanding the theoretical framework.

Findings

Digit sum generating functions relate to B-ary Lambert series.

Transformations of these series exhibit interesting mathematical properties.

Digit sum Dirichlet series reveal new analytical insights.

Abstract

We connect a primitive operation from arithmetic -- summing the digits of a base-$B$ integer -- to $q$-series and product generating functions analogous to those in partition theory. We find digit sum generating functions to be intertwined with distinctive classes of "$B$-ary" Lambert series, which themselves enjoy nice transformations. We also consider digit sum Dirichlet series.