Improving the Explicit Formula for the Riemann Zeta Zeros Using Nonlinear Corrections

We numerically investigate improvements to the explicit Riemann–von Mangoldt formula for counting the zeros of the Riemann zeta function on the critical line. By introducing a nonlinear correction involving a shifted sine integral, we demonstrate significant reduction in maximum deviations from the true counting function. Numerical experiments across extensive ranges of zeros confirm robust improvements and hint at the presence of a sublinear saturation bound, potentially linked to spectral barriers in analytic number theory.