The truthfulness of the Riemann Hypothesis has struggled for a hundred of years since the 19th Century
in the year 1859. However, until now, even the most mathematicians still cannot make a proof or disproof
for such a hypothesis that is convincing for everyone. The most difficult part in the proof or disproof of
the hypothesis is the issue of how one may determine if there are no other critical lines like x = 0.5 that
contain the non-trivial zeta zeros in the Critical Strip Region 0 < x < 1. This writer first employs the
method of “Proving by a Contradiction” and assume that rather than Z = Re(ξ(s = 0.5 + Iy)) = 0, there
is also an additional Z = Re(ξ(s” = x” + Iy)) = 0. However, with the help of a computer software named
“Mathematica” program segment code:
“Plot[Evaluate[ReIm[Zeta[0.5 + I t]]], {t, 0, 30}]” which is just the line
Z = Re(ξ(s”)) = 0, without any other line equal to zero. Obviously, Z = Re(ξ(s = x + Iy)) = 0 at x = 0.5
leads to a contradiction immediately with the assumption that there should be another x” not equal to 0.5
but Z = Re(ξ(s”)) = 0. This result implies that x = 0.5 is the only critical line at x = 0.5. Moreover, for all
the other roots of, this writer has found an ε-δ relationship between Z = Re(ξ(s+/-δ)) = +/-εZ where s =
x + Iy which is sandwiched for a convergence to the Z = 0. There is also another proof by contradiction
to show that there is one and only one critical line for x = 0.5. Therefore, both of these contradictory
proofs show that there is only one critical line at x = 0.5, implying that the Riemann Hypothesis must be
true or the critical line x = 0.5 must contain all of the non-trivial Riemann Zeta zeros.
