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Research Article Open access CC BY 4.0

An Elementary Proof of Riemann’s Hypothesis by the Modified Chi-square Function

Daniele Lattanzi

Journal of Advances in Mathematics and Computer Science · pp. 1–14 · Published 5 Jul 2016

10.9734/BJMCS/2016/25419

Abstract

Riemann’s hypothesis has always been a challenge in number theory, thus the present article shows a general (i.e. valid for any non-trivial zero) and elementary (i.e. not using the theory of complex functions) proof of it, in which the constant +1/2 arises by itself and automatically. In addition the method gives the values of all the trivial zeroes of Riemann’s function. The following steps are used: The modified chi-square function with its parameters Ω, k and ω=ω(k), in one of its four forms (±1∙/)Χk2(Ω,x/ω), as the interpolating function of the {nα} progressions and of the sums {∑nα} with α∈R so that k=2±2α for α<0 and α>0 respectively; the Euler-MacLaurin formula; the shift real vector operator Σ≡ (Σα Σk)≡ (Δα Δk)≡ (+1 (4α–2)) in the Euclidean 2D space (α k) and its extrusion to the imaginary axis it, leading to the 3D shift complex vector operator Σ≡(Σσ Σk Σit)≡(Δσ Δk iΔt)≡(+1 (4σ–2)  it) with norm Σ2=16σ2–16σ+5+t2. The condition Σ=0 that is │Σ│=Σ=0 leads to prove RH.

Riemann’s hypothesis numeric progressions modified chi-square function umbral calculus

References (18)

  1. 1 The Riemann Hypothesis [DOI]
  2. 2 Colloquium : Physics of the Riemann hypothesis [DOI]
  3. 3 Statistical Treatment of Experimental Data [DOI]
  4. 4 Finite operator calculus
  5. 5 H=xp Model Revisited and the Riemann Zeros [DOI]
  6. 6 Distribution of Resonance Widths and Dynamics of Continuum Coupling [DOI]
  7. 7 Nuclei, Primes and the Random Matrix Connection [DOI]
  8. 8 Quantum mechanical potentials related to the prime numbers and Riemann zeros [DOI]
  9. 9 Sequential Representation of Delta Operator in Finite Operator Calculus [DOI]
  10. 10 An elementary and real approach to values of the Riemann zeta function [DOI]
  11. 11 On an expansion theorem in the finite operator calculus of G-C Rota
  12. 12 On p-adic stochastic dynamics, supersymmetry and the Riemann conjecture [DOI]
  13. 13 Scale Laws of Prime Number Frequencies by the Modified Chi-square Function [DOI]
  14. 14 Riemann Zeta Function and Hydrogen Spectrum
  15. 15 Quantization of the Riemann Zeta-Function and Cosmology
  16. 16 QUANTUM HAMILTONIANS AND PRIME NUMBERS
  17. 17 Jost function, prime numbers and Riemann zeta function [DOI]
  18. 18 Proof of the Riemann's hypothesis [DOI]

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