 Solution

Let $L=\log x$ and choosewith fixed positive $a,b$ chosen so that the rectangle up to height $T$ lies inside the Zero-free region of the Riemann zeta function. Move the Perron contour from $1+1/L$ to ${\sigma }_0$. The only singularity crossed is the simple pole at $s=1$ of $-{\zeta }^{\prime }(s)/\zeta (s)$, whose residue contributes $x$.

The standard bound ${\zeta }^{\prime }(s)/\zeta (s)\ll (\log T{)}^2$ in this zero-free rectangle givesThe two horizontal sides are $\ll x(\log T{)}^2/T$, and the truncation error from part a is $\ll x{L}^2/T$. Polynomial factors in $L$ can be absorbed by slightly reducing the exponential constant. Thus some $c>0$ satisfiesThis is the Prime number theorem with classical zero-free-region error.

Solved by gpt-5.6-sol high.