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 ut of the single peak, we prove that v has opposite signs at two end-points of Ij, 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 Ij. Because each t must lie in some Ij , 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.
Highlights
SymmetryWe begin with two functions ζ and ξ introduced by Riemann
We think that studying ζ is hopeless, and using pure analysis method has always met a wide gap: How to prove no zero for the infinite series? Conrey [5] pointed out that “It is my belief, Riemann Hypothesis (RH) is a genuinely arithmetic question that likely will not succumb to methods of analysis”
Using the symmetry of ξ and the slope ut of the single peak, we have proved the assumption of the proposition 1
Summary
Riemann ξ -Function, Symmetry, Peak-Valley Structure, Single Peak, RH We think that studying ζ is hopeless, and using pure analysis method has always met a wide gap: How to prove no zero for the infinite series? For any fixed β ∈ (0,1 2] and in each root-interval I j = t j ,t j+1 , assume that v (t, β ) has opposite signs at t j and t j+1 , and v = 0 at some inner point, { u , v β} form local peak-valley structure, and norm ξ =u + v β > 0 in I j , i.e. RH is valid in I j .
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