What question did this study set out to answer?

The aim is to provide a complete proof of the Riemann Hypothesis using a sieve framework.

May 29, 2026Open Access

Proof of the Riemann Hypothesis Via the CRT Product Structure Sieve Framework

Key Points

The aim is to provide a complete proof of the Riemann Hypothesis using a sieve framework.
Utilized the CRT product structure sieve framework for analysis.
Implemented a rigorous exponential sum cancellation estimate to control error.
Created a hierarchical covering of intervals to establish a global error bound.
Established the relationship π(X)−Li(X)=O(√X log^A X) for the Riemann Hypothesis proof.
Achieved a rigorous control of the sieve error across the parameter space.
Provided a self-contained proof dependent on foundational sieve results.

Abstract

We present a complete proof of the Riemann Hypothesis, specifically the von Koch equivalent form π (X) −Li (X) =O (√X logA X). The proof builds on the CRT product structure sieve framework developed in the companion paper, which provides a linear sieve that circumvents the classical parity problem. The key new ingredient is a rigorous exponential sum cancellation estimate that controls the total sieve error over the parameter space. This estimate, combined with a hierarchical covering of 2, X by intervals of the form (p, p²), yields the required global error bound. The proof is self-contained modulo the foundational results from the sieve framework, which are summarized in Section 2 with complete references.

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