Geometric Proof of Riemann Conjecture (Continued)

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 t j 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 ' ± i β '’ , β '>0 , R 2 =t' 2 + β ' 2 , by symmetry ξ ( τ )= ξ (- τ ) , then -( t ' ± i β '' ) 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.