Twisted Eisenstein series, cotangent‐zeta sums, and quantum modular forms

Abstract

We define twisted Eisenstein series E ± s (h, k; τ) for s ∈ C, and show how their associated period functions, initially defined on the upper half complex plane H, have analytic continuation to all of C := C \ R 0. We also use this result, as well as properties of various zeta functions, to show that certain cotangent-zeta sums behave like quantum modular forms of (complex) weight s.

Eichler integrals of Eisenstein series as $q$-brackets of weighted $t$-hook functions on partitions

“…Remark. This new kind of quantum modularity was also noted by Bettin and Conrey in [3] where they computed E k (z) − z −k E k (− 1 z ) for any k ∈ C. Folsom has recently extended their work in [7], where it was shown that a new family of "twisted Eisenstein series" are holomorphic quantum modular forms, which were then used to show that certain cotangentzeta sums are quantum modular forms in the original sense. Zagier's presentation in [19] shows that for any γ = ( a b c d ) ∈ SL 2 (Z) that h E k ,γ (z) extends to a holomorphic function on the cut plane…”