On an incomplete argument of Erdos on the irrationality of Lambert series

TL;DR

This paper completes Erdos's incomplete proof by demonstrating that Lambert series evaluated at 1/b for negative integers b less than -1 are irrational, using an elementary approach.

Contribution

It provides the first complete elementary proof of the irrationality of Lambert series at specific points, resolving a long-standing incomplete argument by Erdos.

Findings

Lambert series at 1/b for negative integers b < -1 are irrational.

Elementary proof technique established for this irrationality.

Completes Erdos's original incomplete proof.

Abstract

We show that the Lambert series $f(x)=\sum d(n) x^n$ is irrational at $x=1/b$ for negative integers $b < -1$ using an elementary proof that finishes an incomplete proof of Erdos.