Notes on the Riemann \(\zeta\)-function. II
For Part 1, cf. ibid. 139, 310-321 (1998; Zbl 0920.11062). The authors prove that \[ {1\over 2\pi}\int_{\text{Re } s=1/2}{\log|\zeta(s)|\over|s|^2} |ds|= \sum_{\text{Re } \rho>1/2}\log\left|{\rho\over 1-\rho}\right|,\leqno(1) \] where \(\sum_{\text{Re } \rho>1/2}\) denotes summation over complex zeros \(\rho\) of \(\zeta(s)\) having real parts \(> 1/2\), where each zero is counted with its respective multiplicity. In particular, it follows that the Riemann hypothesis is true if and only if \[ {1\over 2\pi}\int_{\text{Re } s=1/2}{\log|\zeta(s)|\over|s|^2} |ds|=0. \] The method of proof of this result belongs to functional analysis, being based on properties of Hardy's space \(H^p({\mathcal D})\), where \({\mathcal D}\) is the unit disk. Jensen's classical formula (\(f(0) \not= 0, r<1\)) \[ {1\over 2\pi}\int_{-\pi}^\pi\log|f(re^{i\theta})|d\theta = \log|f(0)|+ \sum_{|\alpha|<r,f(\alpha)=0}\log{r\over|\alpha|} \] is transformed by the factorization theorem for functions from \(H^p\) when \(r\to 1-0\) in case when \(f(z) = (s-1)\zeta(s)\). After some manipulations, this leads to (1). For Part III, cf. ibid. 149, No. 1, 130-144 (1999).
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