Local Geometric Proof of Riemann Conjecture

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DOI: 10.4236/apm.2020.1010036    452 Downloads   1,404 Views  Citations
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ABSTRACT

Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying ζ and using analysis method likely are two incor-rect guides. Actually, a unique hope may study Riemann function , ,  by geometric analysis, which has the symmetry: v=0 if β=0, and basic expression . We show that |u| is single peak in each root-interval  of u for fixed β ∈(0,1/2]. Using the slope ut, we prove that v has opposite signs at two end-points of Ij. There surely exists an inner point such that , so {|u|,|v|/β} form a local peak-valley structure, and have positive lower bound  in Ij. Because each t must lie in some Ij, then ||ξ|| > 0 is valid for any t (i.e. RH is true). Using the positivity  of Lagarias (1999), we show the strict monotone  for β > β0 ≥ 0 , and the peak-valley structure is equiva-lent to RH, which may be the geometric model expected by Bombieri (2000). This research follows Liuhui’s methodology: “Computing can detect the un-known and method”.

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Chen, C. (2020) Local Geometric Proof of Riemann Conjecture. Advances in Pure Mathematics, 10, 589-610. doi: 10.4236/apm.2020.1010036.

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