## Abstract

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.

Original language | English |
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Pages (from-to) | 1-8 |

Number of pages | 8 |

Journal | Mathematical Physics Electronic Journal |

Volume | 4 |

State | Published - 1 Dec 1998 |

## Keywords

- Dynamical approach
- Maass wave form
- Modular forms
- Modular group
- Period function
- Selberg's zeta function
- Transfer operator