Summa characteristica and the Riemann hypothesis: scaling Riemann’s mountain

Abstract

To prove: Riemann’s hypothesis that all zeta zeros have real part one half. Premise one : the (characteristic) harmonic triangle of Pascal is characterized by the harmonic series. Premise two: when s equals one in the zeta function, the function is identical with the harmonic series. Three : when s equals any complex number, the zeta function is the same as the complex number plane. Four: since the harmonic pattern of the series in the triangle of Pascal completely carries over into the zeta function, and the zeta function charts the complete complex plane (premises one through three combined), whatever places restrictions on the Pascal harmonic pattern also places constraints on the number pattern of the complex plane. Five : the defining (characteristic) restriction responsible for the generation of the whole triangle-plane is the number of both the binomial theorem and the geometrical expansion, which is one half. Because this fraction-interval is recursively present throughout the entire harmonic series, it ties all of the zeta zeros to the real line one half. By recursion we can follow one half as the real part of the zeta zeros forward throughout the entire harmonic series and onto the complex plane. Conclusion: all zeta zeros have real part one half. Since the premises are all true, the conclusion must also be true. Therefore, the Riemann hypothesis is correct. Nature is the realization of the simplest conceivable mathematical ideas. A physicist’s greatest challenge is to uncover the fundamental and universal laws from which a process of simple deduction can lead to a picture of the world. [.] This is what Leibniz felicitously called a ‘re-established’ harmony. Albert Einstein, 1918