I am investigating whether or not there exist $\epsilon > 0$ such that $\zeta(s) \neq 0$ on the strip $1-\epsilon < \Re(s) \leq 1$.
Suppose not. Then given $\delta > 0$ there exists a zero of zeta $\rho$ such that $1 -\delta < \Re(\rho) < 1 $. Hence, there exists a sequence of zeros $\{ z_n \}_{n=1}^\infty$ with increasing imaginary parts that have real part getting closer and closer to the line $\Re(z)=1$.
Can we somehow estimate the density of this infinite set of "fake zeros." The idea is to compare the density of this set with the best known estimate for the density of zeros that are on the critical line.
Best Answer
There are, provably, very few zeros with real part close to $1$ (or bigger than $0.51$ for that matter). These theorems go under the name of "zero density estimates", and they have a vast literature. See Chapter 11 in Ivić: The Riemann zeta function (1985). The book was reprinted in 2013 by Dover Publications.