Author(s)

Chuanmiao Chen^1,2

¹School of Mathematics and Statistics, Central South University, Changsha, China.
²College of Mathematics and Statistics, Hunan Normal University, Changsha, China.

This paper proves Riemann conjecture (RH), i.e., that all the zeros in critical region of Riemann ξ -function lie on symmetric line σ =1/2 . Its proof is based on two important properties: the symmetry and alternative oscillation for ξ = u + iv . Denote . Riemann proved that u is real and v ≡ 0 for β =0 (the symmetry). We prove that the zeros of u and v for β > 0 are alternative, so u (t,0) is the single peak. A geometric model was proposed. is called the root-interval of u (t,β) , if |u| > 0 is inside I_j and u = 0 is at its two ends. If |u (t,β)| has only one peak on each I_j, which is called the single peak, else called multiple peaks (it will be proved that the multiple peaks do not exist). The important expressions of u and v for β > 0 were derived. By , the peak u (t,β) will develop toward its convex direction. Besides, u_t (t,β) has opposite signs at two ends t = t_j , t_j+1 of I_j , also does, then there exists some inner point t′ such that v (t′,β) = 0. Therefore {|u|,|v|/β} in I_j form a peak-valley structure such that has positive lower bound independent of t ∈ I_j (i.e. RH holds in I_j ). As u (t,β) does not have the finite condensation point (unless u = const.), any finite t surely falls in some I_j , then holds for any t (RH is proved). Our previous paper “Local geometric proof of Riemann conjecture” (APM, V.10:8, 2020) has two defects, this paper has amended these defects and given a complete proof of RH.

Riemann Conjecture, Geometric Analysis, Symmetry, Alternative Oscillation, Single Peak, Peak-Valley Structure

Chen, C. (2021) Geometric Proof of Riemann Conjecture. Advances in Pure Mathematics, 11, 334-345. doi: 10.4236/apm.2021.114021.