A Carleman-Nevanlinna theorem and summation of the Riemann zeta-function logarithm
In this paper the authors proves a Carleman-Nevanlinna theorem for a rectangle. This result is is applied to the summation of \(\log| \zeta(s)| \) on the critical and other vertical lines, where \(\zeta(s)\) means the Riemann zeta-function. In particular, let \[ I(\varepsilon)=\int_0^\infty e^{-\varepsilon t}\log\left| \zeta\left(\frac{1}{2}+it\right) \right|\, dt,\, \varepsilon>0, \] and let \(\{\rho_j\}\) be the non-trivial zeros of \(\zeta(s)\). Then \[ \frac{\pi}{2}\sum_{j}\left| \Re \rho_j-\frac{1}{2}\right| =I(+0)+\frac{\pi}{2}, \] where \(I(+0):=\lim_{\varepsilon\to 0}I(\varepsilon)\). Hence, the Riemann hypothesis for \(\zeta(s)\) holds if and only if \(I(+0)=-\pi/2\).
scientific article; zbMATH DE number 2186786
scientific article