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

Cheng-Hung Chang*, Dieter Mayer

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


We calculate the period function of Lewis of the automorphic Eisenstein series E(s, w) = 1/2vsn,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)ψs(z), where B(x, y) denotes the beta function and ψs a function introduced some time ago by Zagier and given for Rs > 1 by the series ψs(z) = ∑n,m≥1(mz + n)-2s + 1/2ζ(2s) (1 + z -2s). The analytic extension of ψs 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 languageEnglish
Pages (from-to)1-8
Number of pages8
JournalMathematical Physics Electronic Journal
StatePublished - 1 Dec 1998


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


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