Some identities of Ramanujan type

We will discuss some algebraically independent functions connected to Eisenstein series and having degeneracy of the transcendence degree of the field generated by their values. This degeneracy is a consequence of some quasi- modular identities that generalize the classical identities of S. Ramanujan and E. Grosswald. Values of these functions at the point i are connected to Riemann zeta-values. For example the following classical equality of M. Lerch holds
$$ \zeta(3)=\frac{7\pi^3}{180}-2\sum_{n=1}^\infty\sigma_{-3}(n)e^{-2\pi n}, \qquad \sigma_{-3}(n)=\sum_{d|n}d^{-3}. $$