Prime Residual Amplification: A New Asymptotic Strategy for the Riemann Hypothesis

The Asymptotic Limit of Prime Residuals

Research presented in hal-01562597 establishes a significant limit concerning the nth prime number, p_n. The core finding demonstrates that the ratio of the index n to the residual p_n - n(log n + log(log n) - 1) diverges to infinity. This result provides a strategic framework for approaching the Riemann Hypothesis by quantifying the error terms in prime distribution and mapping them to the behavior of the Riemann zeta function.

The Epsilon-Convergence Framework

The framework utilizes an epsilon function, epsilon(n), to track the rate at which the nth prime approaches its asymptotic value. For n greater than or equal to 688383, the paper provides a concrete upper bound: p_n is less than or equal to n multiplied by the sum of log n, log(log n), -1, and the inverse of epsilon(n).

• Theoretical Link: If the Riemann Hypothesis is true, the growth of epsilon(n) must be strictly controlled. Any deviation in the divergence of this ratio would imply the existence of zeros of the Riemann zeta function away from the critical line.
• Zero-Free Regions: The divergence rate of epsilon(n) corresponds to the width of the zero-free region. Sharpening the lower bound of epsilon(n) directly constricts the possible locations of non-trivial zeros in the critical strip.

The Amplifier Criterion: A Pathway to Proof

A novel research approach involves the Amplifier Criterion, which defines the ratio A(n) = n / (p_n - n(log n + log(log n) - 1)). This methodology transforms the Riemann Hypothesis into a lower-bound problem for A(n).

Proposed Theorem: The Riemann Hypothesis is equivalent to the assertion that the error term of the nth prime is bounded by the square root of n times a poly-logarithmic factor. This frames the proof as a problem of establishing a specific growth rate for A(n) that is consistent with the square-root cancellation predicted by the critical line distribution.

Implementation and Computational Validation

The following Wolfram code allows for the empirical analysis of the amplifier's growth across large sequences of primes, providing a tool for testing the conjectures derived from the paper hal-01562597.

DaoudiRatio[n_] := n / (Prime[n] - n * (Log[n] + Log[Log[n]] - 1)); ListLinePlot[Table[DaoudiRatio[n], {n, 1000, 1000000, 1000}], PlotRange -> All, PlotLabel -> "Growth of Prime Amplifier"]

Tangential Connections: Spectral Theory and Prime Gaps

The distribution of these residuals may be linked to the eigenvalue spacing of random matrices. The divergence observed in the study suggests that the local density of primes, when normalized by the Daoudi limit, follows patterns found in the Gaussian Unitary Ensemble (GUE). This provides a bridge between analytic number theory and arithmetic quantum chaos, suggesting that prime residuals are governed by a spectral density symmetric around the critical line.