Abstract
Proposed by Bernhard Riemann in 1859, Riemann hypothesis refers to the famous conjecture explicitly equivalent to the mathematical statement that the critical line in the critical strip of Riemann zeta function is the location for all non-trivial zeros. The Dirichlet eta function is the proxy for Riemann 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 Riemann-Dirichlet Ratio). We recognize and propose the Sigma-power laws (in both the Dirichlet and Riemann versions) and the Riemann-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 Riemann 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.
Highlights
In 1859, the famous German mathematician Bernhard Riemann proposed the Riemann hypothesis
Proposed by Bernhard Riemann in 1859, Riemann hypothesis refers to the famous conjecture explicitly equivalent to the mathematical statement that the critical line in the critical strip of Riemann zeta function is the location for all non-trivial zeros
We discovered our key formula which is based on our key Ratio
Summary
In 1859, the famous German mathematician Bernhard Riemann proposed the Riemann hypothesis. The official statement of Riemann hypothesis as a Millennium Prize Problem was given by mathematician Enrico Bombieri. Riemann zeta function is the location for all of its non-trivial zeros. The critical strip of Riemann zeta (ζ) function can be represented by the one and only one critical line in the middle and an infinite number of other parallel lines on either side of this critical line, with every single line mathematically described by this function with a particular designated sigma (σ) value comprising of real numbers. Theorem I: The exact same Riemann-Dirichlet Ratio, directly derived from either the Riemann zeta or Dirichlet eta function, is an irrefutably accurate mathematical expression on the de novo criteria for the actual presence [but not the actual locations] of the complete set of (identical) infinite non-trivial zeros in both functions.
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