Proofs for two Lambert series identities of Gosper (Q2050203)

In this paper, the author proves two of the three identities given in 2001 by \textit{R. W. Gosper} [Dev. Math. 4, 79--105 (2001; Zbl 1058.33017)] as identities in a conjecture related to Lambert series. The other one was recently proven in [\textit{M. El Bachraoui}, Proc. Am. Math. Soc. 147, No. 10, 4451--4464 (2019; Zbl 1469.33013)]. The methods used here are the ones from discrete group theory, modular group and its congruence subgroups, properties of Eisenstein series and comparing coefficients.