Automorphy of the principal Eisenstein series of weight 1: an application of the double sine function (Q1047759)

For \(n \in \mathbb{N}\), \(k \in \mathbb{C}\) and \(\tau \in \mathbb{C}\) such that \(\mathrm{Im} (\tau) >0\) let \[ E_k(\tau)= \frac{\zeta(1-k)}{2} + \sum\limits_{n=1}^{\infty} \sigma_{k-1}(n) \exp(2\pi i n \tau), \] where \(\sigma_k(n) := \sum\limits_{d\mid n} d^k\). For an even integer \(k\geq 4\), \(E_k(\tau)\) is the Eisenstein series of weight \(k\) for the full modular group, hence satisfies the automorphy condition \[ E_k(-1/\tau) = \tau^k E_k(\tau). \] Let \[ R_k(\tau):=E_k(-1/\tau) - \tau^k E_k(\tau). \] Then, \(R_k(\tau)=0\) for all even integers \(k\geq 4\) and \(\tau \in \mathbb{C}\) such that \(\mathrm{Im} (\tau) >0\). Furthermore, the transformation formula for the Dedekind \(\eta-\)function implies that \(R_2(\tau)=-\tau/(4 \pi i)\). In the paper under review, boundary values of the function \(R_1 (\tau)\) as \(\tau\) approaches the real axes are related to certain special values of (derivatives of) the double sine function. The main result of the paper, stating that \[ \lim \limits_{\tau \rightarrow n}R_1(\tau) = \frac{1}{2i} \left( \frac{1}{n} \sum\limits_{k=1}^{\left\lfloor n/2 \right\rfloor}(n-2k) \cot(\pi k/n) -\frac{1}{\pi} \right) \] and \[ \lim \limits_{\tau \rightarrow 1/n}R_1(\tau) = \frac{1}{2ni} \left( \frac{1}{n} \sum\limits_{k=1}^{\left\lfloor n/2 \right\rfloor}(n-2k) \cot(\pi k/n) -\frac{1}{\pi} \right) \] for a positive integer \(n\) is deduced from the properties of the double sine function.