Formal group laws for multiple sine functions and applications (Q1951667)

Let \[ E_1(\tau)=-\frac{1}{4} +\sum_{n=1}^{\infty} d(n)q^n=-\frac{1}{4}+\sum_{m,n \geq 1} q^{mn}, \] where \(q=\exp(2\pi i \tau)\) and \(\tau\) belongs to the upper half-plane. Suppose \(M,N \geq 1\) are two coprime integers. The authors prove that the limit \(\lim_{\tau \to M/N} (E_{1}(-1/\tau)-\tau E_1(\tau))\) is equal to \[ \frac{i}{2N} \sum_{m,n} \Big|\frac{m}{M}-\frac{n}{N} \Big| \cot \Big(\pi \Big|\frac{m}{M}-\frac{n}{N} \Big| \Big) +\frac{i}{2\pi N}, \] where the sum is taken over every pair of integers \((m,n)\) satisfying \(0 \leq m<M\) and \(0 \leq n <N \) except for the pair \((0,0)\).