We consider the classical Chazy equation, which is known to be integrable in hypergeometric functions. But this solution has remained purely existential and was never used numerically. We give explicit formulas for hypergeometric solutions in terms of initial data. We have found a special solution of this type in the upper half plane H with the same tessellation of H as that of the modular group. This
allowed us to derive some new identities for the Eisenstein series. We constructed a special solution in the unit disk and gave an explicit description of singularities on its natural boundary. Finally, we found an explicit global solution to Chazy equation
in elliptic and theta functions. The results have some applications to analytic number theory.

Keywords:

Chazy equation, hypergeometric parametrization, modular group, Eisenstein series, sum of divisors, Riemann hypothesis