TITLE:
Geometric Proof of Riemann Conjecture (Continued)
AUTHORS:
Chuanmiao Chen
KEYWORDS:
RC, Equivalence, RC2, Product Expression, Single Peak, Multiple Zeros
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.11 No.9,
September
22,
2021
ABSTRACT: This paper will prove Riemann conjecture(RC): All zeros of ξ(τ) lie on critical line. Denote ,and on critical line. We have found two mysteries in Riemann’s paper. The first mystery is the equivalence: is uniquely determined by its initial value u (t). The second mystery is Riemamm conjecture 2 (RC2): Using all zeros tj of u (t) can uniquely express .We find that the proof of RC is hidden in it. Our basic idea as follows. Consider functional equation . It is known that on critical line and , then we have the upper bound of growth To prove RC2 (or RC), by contradiction. If ξ(τ) has conjugate complex roots t'±iβ'’, β'>0, R2=t'2+β'2, by symmetry ξ(τ)=ξ(-τ), then -(t'±iβ'') do yet. So ξ must contain four factors. Then u(t) contains a real factor and ln|u(t)| contains a term (the lower bound) which contradicts to the growth above. So ξ can not have the complex roots and u(t) does not have the factor p(t). Therefore both RC2 and RC are proved. We have seen that the two-dimensional problem is reduced to one-dimension and the one-dimensional u(t) is reduced to its product expression. Perhaps this is close to the original idea of Riemann. Other results are also discussed by geometric analysis in the last section.