Computational Methods and Function Theory 4 (2004), No. 2, 391--403
Copyright Heldermann Verlag 2004
Andriy A. Kondratyuk
kond@franko.lviv.ua , Lviv National University, Department of Mathematics, Universitetska Str. 1, 79000 Lviv, Ukraine.
A Carleman-Nevanlinna Theorem for a rectangle is proved. The theorem is applied to the summation of $\log|\zeta(s)|$ on the critical and other vertical lines, where $\zeta(s)$ is the Riemann zeta-function. In particular, let $$ I(\varepsilon)= \int_0^\infty e^{-\varepsilon t} \log\!\left|\zeta\!\left(\frac12+it\right)\right|\,dt, \qquad \varepsilon>0, $$ and let $\left\{\rho_j\right\}$ be non-trivial zeros of $\zeta(s)$, then $$ \frac\pi2\sum_j\left|\Re\rho_j-\frac12\right|=I(+0)+\frac\pi2, $$ where $I(+0):=\lim_{\varepsilon\to0}I(\varepsilon)$. Thus, the Riemann hypothesis for $\zeta(s)$ holds if and only if $I(+0)=-\pi/2$.
Keywords: Meromorphic function, Carleman-Nevanlinna Theorem, summation, Riemann zeta-function, Euler product, Riemann hypothesis, Dirichlet series, Selberg class, $L$-function.
MSC 2000: Primary 30Dxx, 11M06; Secondary 11M41.
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