Number Theory

From Ramanujan to Riemann Hypothesis

Authors: Payam Danesh, Raoul Bianchetti
Comments: 16 Pages.

In this work we offer a careful framework for approaching the critical-line problem associated with the Riemann zeta function. At its heart is a long-standing divide in the subject. On one side are analytic approaches, which study the completed zeta function through its reflection symmetry. On the other side are arithmetic approaches, where related criteria often appear through extreme behavior in divisor functions. The purpose of this paper is not to claim a proof of the Riemann Hypothesis, but to place these two perspectives into a clearer and more usable relationship. The argument begins with reflected analytic data for the completed zeta function. It shows that such data can be described through an odd analytic perturbation, giving a more organized way to understand the analytic side of the problem. This also resolves a common point of confusion: the full complex defect is not required to vanish on the critical line. What matters is more subtle. Under a natural real-symmetry condition, the real part of the defect vanishes on the critical line, and this is the feature that becomes useful for the bridge argument. The arithmetic side is built around Ramanujan’s logarithmic divisor profile. The paper establishes the existence and positivity of the relevant extreme scale in the range needed for the proposed connection. These analytic and arithmetic pieces are then brought together through a real bridge functional, made up of a main sign term and a correction term. The main outcome is a conditional criterion for the critical line. If the bridge functional is zero-adapted at the nontrivial zeros, if the real analytic defect satisfies the required one-sided sign condition, and if the correction term remains strictly smaller than the main term, then every nontrivial zero must lie on the critical line. The contribution of this work is therefore structural rather than conclusive. It does not present the Riemann Hypothesis as solved. Instead, it separates what is already established from what still needs to be proved. The key sign law, the domination estimate, and the zero-adaptation identity remain open requirements for any future application of the framework. Its practical value is that it gives researchers a precise checklist for testing whether a proposed analytic or arithmetic strategy can genuinely support a critical-line argument.
Category: Number Theory