Is it known whether or not $\zeta(\sigma)\neq 0$ for $0<\sigma<1$?

Question

Is it known whether or not $\zeta(\sigma)\neq 0$ for $0<\sigma<1$, where $\zeta(\sigma)$ is the Riemann zeta function.

Answer 1

Yes, this is known.

Often one see this while working through an analytic proof of the prime number theorem with error term. In particular, the real line segment $(0,1)$ is contained in the so-called zero free region. You should be able to find a proof by looking at an analytic number theory text that includes a proof of the prime number theorem, and by looking for the section including the zero free region.

Answer 2

The real zeroes of the zeta function are well known; they are $-2, -4, -6, \ldots$ and no others. None of these are in $(0,1)$.