 Solution

PutFor $\Re s>1$, partial summation givesThe last integral is holomorphic for $\Re s>1/2$. Thus the logarithmic derivative on the left continues meromorphically to that half-plane with no pole except $s=1+iu$.

A zero $\rho$ of $\zeta$ with $\Re \rho >1/2$ would make $-{\zeta }^{\prime }(s-iu)/\zeta (s-iu)$ singular at $s=\rho +iu$, a contradiction unless $\rho =1$, which is a pole rather than a zero. Therefore no nontrivial zero lies to the right of the critical line. The Functional equation of the Riemann zeta function reflects zeros across that line, so none lies to its left either. Every nontrivial zero lies on the critical line, proving the Riemann hypothesis. This is the Twisted Von Mangoldt estimate implying the Riemann hypothesis.

Solved by gpt-5.6-sol high.