Spectral Rigidity and the Explicit Formula - An RMT Perspective on Arithmetic Reconstruction

The explicit formulas of Riemann and von Mangoldt connect the distribution of prime numbers to the zeros of the Riemann zeta function. While this duality is well-known, the mechanism by which the oscillatory sum of zeros converges to a stable arithmetic signal remains non-trivial from a spectral analysis perspective. This paper investigates the role of spectral rigidity in the inverse reconstruction of the von Mangoldt function. Explicitly assuming the Riemann Hypothesis and Montgomery's Pair Correlation Conjecture, we analytically derive that the variance of the smoothed reconstruction operator is governed by the spectral structure factor.