On the zeros of the Riemann zeta function along the critical line

By Eric A. Galapon | May 3, 2025

If $s$ is a real sumber such that

(1) $\begin{equation*} \int_0^{\infty} \frac{\sinh t - t \cosh t}{t^{\frac{1}{2}+i s} \sinh^2 t}\,\mathrm{d}t = 0, \end{equation*}$

then

(2) $\begin{equation*} \zeta\!\left(\frac{1}{2}+i s\right) = 0 . \end{equation*}$