The Symmetry of Riemann <i>ξ</i>-Function

Abstract

To prove RH, studying ζ and using pure analysis method likely are two kinds of the incorrect guide. Actually, a unique hope may study Riemann function by geometric analysis, which has the symmetry: v = 0 if β = 0, and Assume that | u | is single peak in each root-interval of u for any fixed β ∈ (0,1/2] , using the slope u t of the single peak, we prove that v has opposite signs at two end-points of I j , there surely is an inner point so that v = 0, so {| u |,| v |/ β }form a local peak-valley structure, and have positive lower bound in I j . Because each t must lie in some I j , then || ξ || > 0 is valid for any t . In this way, the summation process of ξ is avoided. We have proved the main theorem: Assume that u ( t , β ) is single peak, then RH is valid for any . If using the equivalence of Lagarias (1999), the assumption of single peak can be canceled. Therefore our new thinking is that we have found the local peak-valley structure of ξ , which may be the geometry structure expected by Bombieri (2000), and proposed a basic framework of proving RH by geometric analysis.