Abstract
We investigate a proposed criterion for the Riemann Hypothesis (RH) based on the discrepancy between the Riemann zeta function and a corrected finite Euler product. Let Z_X(s) = ζ(s)/E*_X(s), where E*_X is a suitably defined corrected Euler approximation. For a fixed point s₀ with Re(s₀) > 1/2, define the Euler–Cáceres gap as one half of the logarithm of the Poisson-weighted boundary mean square of Z_X on the critical line, minus log|Z_X(s₀)|. The proposed equivalence states that RH holds if and only if this gap is O(1/log X) at a single fixed point. A related proposed criterion asks whether the weighted mean square of Z_X on the critical line has a finite limit inferior; under RH, the claimed limit is π. These equivalences require verification under the exact definition and analytic properties of E*_X. The central mechanism is a Poisson–Jensen decomposition: zeros to the right of the critical line can contribute positive Green-function terms, while the remaining part of the gap arises from boundary averaging. This suggests that decay of the gap at one interior point could exclude all off-line zeros. A rigorous treatment must account for the pole of ζ at s = 1, the convergence of zero sums, boundary behavior, and possible contributions at infinity. We propose a three-stage research program. First, derive and validate the exact decomposition at the convenient point s₀ = 2. Second, investigate arithmetic cancellation in the Euler-product remainder and seek to improve the currently stated bound B_X ≪ √X/log X to the intermediate estimate B_X ≪ε X^ε. Third, seek the decisive logarithmic estimate B_X = O(1/log X) without assuming RH. Numerical tests and comparisons with functions possessing off-critical-line zeros may guide the analysis but cannot replace an unconditional proof.
Keywords: Riemann Hypothesis; Euler products; Poisson–Jensen formula; mean-square estimates; zeta zeros; arithmetic cancellation.