Sometime ago, I read somewhere (should be in Titschmarsh) that, if $N(\sigma, T)$ denotes the number of zeros of the Riemann zeta function $\zeta(s)$ with $\Re(s)\geq \sigma>1/2$ up to height $T>0$, then some result of Ingham says $$N(\sigma, T)\ll T^{\frac{3(1-\sigma)}{2-\sigma}}\log^{5}T.$$

Where can I find a proof of this result? A Google search didn't yield much.

zero-density theorems, if you want to look for further developments. $\endgroup$ – Greg Martin Feb 9 '19 at 0:48