Hilbert transform of \(\text{log}|f|\) (Q2759014)

Let \(f\) be an entire function of the Cartwtight class and \(\tilde{\cdot}\) be the Hilbert transform of \(\cdot\). The author derives the following formula NEWLINE\[NEWLINE\widetilde{\log |f|} (t) = - \pi \nu _{\{x_n\}} (t) + \Biggl(\frac{h_+ + h_-}{2}\Biggr) t - \sum _n \varphi _{z_n} (t) + \theta .NEWLINE\]NEWLINE Here \(\{x_n\}\) is the sequence of all real roots of \(f\) and \(\nu \) is their counting function, \(h_{\pm}\) is the indicator function of \(f\) at \(\pm \pi /2\), \(\{z_n\}\) is the sequence of all non-real roots of \(f\), \(\varphi _{z_n} (t)\) is a certain branch of the argument of the Blaschke primary factor corresponding to \(z_n^4,\) and \(\theta\) is a constant.