Limit values of Eisenstein series and multiple cotangent functions
Let \(k\) be a positive integer. Set \[ E_k(\tau) = \frac{\zeta(1-k)}{2} + \sum_{n=1}^{\infty} \sigma_{k-1}(n) e^{2\pi i n \tau}, \] where \(\text{Im}(\tau) > 0\) and \[ \sigma_{k-1}(n) = \sum_{d\mid n} d^{k-1}. \] When \(k\) is even, since \(E_k\) is a modular form, \[ E_k\left( -\frac{1}{\tau} \right) - t^k E_k(\tau) = 0 \quad(\tau \in \mathbb{H}). \] However, when \(k\) is odd, the above does not necessarily hold. The main result of the paper is Theorem 1: For each positive integer \(k\) we have: \[ \lim_{\tau \rightarrow 1} \left( E_{k}\left( -\frac{1}{\tau} \right) - t^{k} E_{k}(\tau) \right) = \frac{(-1)^{k}B_{k-1}}{2 \pi i}\tag{1} \] \[ \lim_{\tau \rightarrow 2} \left( E_{k}\left( -\frac{1}{\tau} \right) - t^{k} E_{k}(\tau) \right) = \frac{(-1)^{k}B_{k-1}}{2 \pi i} 2^{k-1}\tag{2} \] \[ \lim_{\tau \rightarrow \frac{1}{2}} \left( E_{k}\left( -\frac{1}{\tau} \right) - t^{k} E_{k}(\tau) \right) = \frac{(-1)^{k}B_{k-1}}{2 \pi i} \frac{1}{2}.\tag{3} \] Here \(B_{n}\) is the usual Bernoulli number. The proof uses the theory of multiple sine functions [see \textit{N. Kurokawa}, Proc. Japan Acad., Ser. A 68, 256--260 (1992; Zbl 0797.11053) and \textit{N. Kurokawa} and \textit{S. Koyama}, Forum Math. 15, No. 6, 839--876 (2003; Zbl 1065.11065)].
scientific article; zbMATH DE number 5285745