In my thesis, i stumbled across the following problem:

Define $$F(\sigma, T)=\frac{1}{T}\int_{-T}^{T} \frac{1}{|\zeta(\sigma+it)|^2} \mathrm{d}T$$ where $\zeta$ denotes the Riemann zeta function.

Is the statement that $\lim_{T\rightarrow \infty} F(\sigma, T)=O(1)$ equivalent to the statement that $\zeta(s)\neq 0$ for $\Re(s)>\sigma$ ?

In my thesis, i stumbled across the following problem:

Define $$F(\sigma, T)=\frac{1}{T}\int_{-T}^{T} \frac{1}{|\zeta(\sigma+it)|^2} \mathrm{d}T$$ where $\zeta$ denotes the Riemann zeta function.

Is the statement that $\lim_{T\rightarrow \infty} F(\sigma, T)=O(1)$ equivalent to the statement that $\zeta(s)\neq 0$ for $\Re(s)\geq\sigma$ ?