TITLE:
Local Geometric Proof of Riemann Conjecture
AUTHORS:
Chuanmiao Chen
KEYWORDS:
Riemann Conjecture, Local Geometric Proof, Symmetry, Peak-Valley Struc-ture, Equivalence, Liuhui’s Methodology
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.10 No.10,
October
19,
2020
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 ofu for fixed β∈(0,1/2]. Using the slope ut, we prove that vhas 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 eacht must lie in some Ij, then ||ξ||>0 is valid for anyt (i.e. RH is true). Using the positivityof 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”.