Abstract: The Riemann-Siegel formula is the main tool for calculating the values of the Riemann zeta function on the critical line, but its complexity makes it difficult to extract theoretical information about the zeros of zeta, a key challenge of the Riemann Hypothesis. This talk introduces an alternative to the Riemann-Siegel formula using higher-order sections, based on our previous results on the approximate functional equation with exponentially decaying error. This alternative defines the N-th variation space of sections of zeta, Z_N(t; a_1,...,a_N), with the following properties:

1. Core Function: At the origin, Z_0(t) has well-understood zeros.
2. Approximation: At Z_N(t;1,...,1), it approximates zeta in the region 2N < t < 2N+2.

This transforms the Riemann Hypothesis into a non-linear optimization problem, studying the dynamic change of zeros along paths from a = (0, ..., 0) to a = (1, ..., 1). The focus will be on the discriminant of multiple zeros forming within this space, revealing a dynamic repulsion relation between consecutive zeros of zeta, deeply related to Gram’s law. No prior knowledge of the zeta function is required.