On a multiple trigonometric series (Q5890473)

The author proves that there exists a real number \(\alpha\) such that the sequence NEWLINE\[NEWLINE \{ h_N(\alpha) \}, \quad \text{where} \quad h_N (\alpha):= \sum_{k=1}^N \sum_{l=1}^N \frac{\sin (\alpha kl)}{kl} NEWLINE\]NEWLINE diverges as \(N \to \infty\). Here, \(\alpha\) is defined by \(\alpha = 2 \pi \sum_{n=1}^{\infty} \frac{1}{q_n}\) , where \(q_1 =2\) and \(q_{n+1} = q_n^{n q_n +1}\) for \(n \geq 1\) .