= Jensen disk proof of the zeta zero-count bound
{c}
{title2=$N(T)\ll T\log T$}
Put $F(s)=(s-1)\zeta(s)$. The <fractional-part continuation formula for the Riemann zeta function> gives $|F(s)|\ll(|s|+1)^2$ for $\operatorname{Re}s\geq1/4$. At $2+ij$, the reciprocal <Euler product> bounds $|\zeta(2+ij)|$ below by $1/\zeta(2)$. Applying the <Jensen zero-count bound> with outer radius $7/4$ and inner radius $8/5$ gives $O(\log(|j|+2))$ zeros in the latter disk. Those disks cover the half-strip $1/2\leq\operatorname{Re}s\leq1$. Reflecting zeros using the <functional equation of the Riemann zeta function> completes the $O(T\log T)$ bound without requiring left-half-plane growth estimates.
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