Proofs for two Lambert series identities of Gosper

TL;DR

This paper uses modular forms and Lambert series techniques to prove and confirm two Lambert series identities conjectured by Gosper, advancing understanding in this area of mathematical analysis.

Contribution

It establishes two Lambert series identities of Gosper using modular forms and Lambert series manipulations, confirming conjectures in the field.

Findings

Confirmed a Lambert series identity conjectured by Gosper

Established an Eisenstein series identity using modular forms

Validated another Lambert series identity of Gosper

Abstract

Applying the theory of modular forms and Lambert series manipulations we establish an Eisenstein series identity. From this formula we confirm a Lambert series identity conjectured by Gosper. Another Lambert series identity of Gosper is also confirmed by using Lambert series manipulations.