Limit formulas for non-modular Eisenstein series (Q2880111)

For \(k\) a positive integer, define the Eisenstein series NEWLINE\[NEWLINEE_{k}(\tau) = \frac{\zeta(1-k)}{2} + \sum_{n=1}^{\infty} \sigma_{k-1}(n) e^{2\pi i n \tau}, NEWLINE\]NEWLINE where \(\tau\) lies in the upper half-plane, and \(\sigma_{k-1}(n) = \sum_{d | n}d^{k-1}.\)NEWLINENEWLINEFor \(k \geq 4, \) even, \(E_{k}\) is a modular form, and hence \(E_{k}(-1/\tau) - \tau^{k}E_{k}(\tau)\) is identically zero. In the paper under review, the author studies the above difference for \(k \geq 3, \) odd. The main result isNEWLINENEWLINE{Theorem 1.} Let \(k\) be an odd integer with \(k \geq 3. \) Then for each real number \(x > 0,\) NEWLINE\[NEWLINE\lim_{\tau \rightarrow x} \left( E_{k}(-1/\tau) - \tau^{k}E_{k}(\tau) \right) = \frac{(k-1)!}{(2 \pi i)^{k}} \left( \zeta(k)(1+x^{k}) + 2x^{k}\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{(mx+n)^{k}} \right).NEWLINE\]