Symmetry of zeros in the critical strip for Riemann Hypothesis

Question

If it can be proven that there are no zeros for real values greater than $0.5$ in the critical strip, does this prove that there are no zeros in the critical strip having a real value of less than $0.5$?

Answer 1

Yes, it would prove that.

The Riemann $\zeta$ function is known to satisfy the following functional equation: $$ \zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}2\right)\Gamma(1-s)\zeta(1-s) $$ Assuming $z_0$ is a root in the critical strip which is not on the critical line, then $1-z_0$ is also a root on the critical strip which is on the opposite side of the critical line. So if $z_0$ has real part less than $0.5$, then $1-z_0$ has real part greater than $0.5$, and vice versa. Of course this also means that the absence of roots to the right of the critical strip imples the absence of roots on the left side.

This phenomenon of roots off the critical line appearing in pairs is actually why the Riemann hypothesis is so important. One manifestation is the following: If $\pi:\Bbb R^+\to \Bbb N$ is the prime counting function (i.e. $\pi(x)$ is the number of primes less than or equal to $x$), then we have $$ \pi(x) = \int_2^x\frac{dx}{\ln x} + \text{correction terms} $$ The corrections include an infinite sum with one term for each non-trivial root of the Riemann $\zeta$ function. If there are roots not on the critical line, then this symmetry means there are actually two roots of that magnitude, meaining we have more, larger correction terms.