The period function of the nonholomorphic Eisenstein series for PSL(2, ℤ)

We calculate the period function of Lewis of the automorphic Eisenstein series E(s, w) = 1/2v^s ∑_n,m≠(0,0)(mw + n) ^-2s for the modular group PSL(2, ℤ). This function turns out to be the function B(1/2, s + 1/2)ψ_s(z), where B(x, y) denotes the beta function and ψ_s a function introduced some time ago by Zagier and given for Rs > 1 by the series ψ_s(z) = ∑_n,m≥1(mz + n)^-2s + 1/2ζ(2s) (1 + z ^-2s). The analytic extension of ψ_s to negative integers s gives just the odd part of the period functions in the Eichler, Shimura, Manin theory for the holomorphic Eisenstein forms of weight -2s + 2. We find this way an interesting connection between holomorphic and nonholomorphic Eisenstein series on the level of their respective period functions.