Rigorous Proof for Reimann Hypothesis Using the Novel Sigma-power Laws and Concepts from the Hybrid Method of Integer Sequence Classification

Abstract

Proposed by Bernhard Reimann in 1859, Reimann hypothesis refers to the famous conjecture explicitly equivalent to the mathematical statement that the critical line in the critical strip of Reimann zeta function is the location for all non-trivial zeros. The Dirichlet eta function is the proxy for Reimann zeta function. We treat and closely analyze both functions as unique mathematical objects looking for key intrinsic properties and behaviors. We discovered our key formula (coined the Sigma-power law) which is based on our key Ratio (coined the Reimann-Dirichlet Ratio). We recognize and propose the Sigma-power laws (in both the Dirichlet and Reimann versions) and the Reimann-Dirichlet Ratio, together with their various underlying mathematically-consistent properties, in providing crucial \textit{de novo} evidences for the most direct, basic and elementary mathematical proof for Reimann hypothesis. This overall proof is succinctly summarized for the reader by the sequential Theorem I to IV in the second paragraph of Introduction section. Concepts from the Hybrid method of Integer Sequence classification are important mathematical tools employed in this paper. We note the intuitively useful mental picture for the idea of the Hybrid integer sequence metaphorically becoming the non-Hybrid integer sequence with certain criteria obtained using Ratio study.