An Expression For The Argument of $\zeta$ at Zeros on the Critical Line

TL;DR

This paper proposes a conjectured explicit formula for the argument of the Riemann zeta function at its zeros on the critical line, suggesting a potential proof of the Riemann Hypothesis through a transcendental equation.

Contribution

It introduces a new conjectured expression for the argument of ζ at its zeros, linking it to a transcendental equation that could prove the Riemann Hypothesis.

Findings

Conjectured explicit formula for the argument of ζ at zeros.

Proposes that solutions to a transcendental equation imply the Riemann Hypothesis.

Links the counting functions of zeros on the critical line and strip.

Abstract

The function $S_n (t) = \pi \left( \frac{3}{2} - {frac} \left( \frac{\vartheta(t)}{\pi} \right) + \left( \lfloor \frac{t \ln \left( \frac{t}{2 \pi e}\right)}{2 \pi} + \frac{7}{8} \rfloor - n \right) \right)$ is conjectured to be equal to $S (t_n)_{} = \arg \zeta \left( \frac{1}{2} + i t_n \right)$ when $t=t_n$ is the imaginary part of the n-th zero of $\zeta$ on the critical line. If $S(t_n)=S_n(t_n)$ then the exact transcendental equation for the Riemann zeros has a solution for each positive integer $n$ which proves that Riemann's hypothesis is true since the counting function for zeros on the critical line is equal to the counting function for zeros on the critical strip if the transcendental equation has a solution for each $n$.