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New expressions are given for the Fourier expansions of non-holomorphic Eisenstein series with weight k. Among other applications, this leads to non-holomorphic analogs of formulas of Ramanujan, Grosswald and Berndt containing Eichler integrals of holomorphic Eisenstein series.

Formulas for non-holomorphic Eisenstein series and for the Riemann zeta function at odd integers