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.