Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 150 / 3 / c

For $\sigma >1$, the three-four-one zero-free-region argument starts fromApplying this termwise togives

Suppose $\rho =\beta +i\gamma$ is a zero with $|\gamma |\ge 4$. In the supplied Local partial-fraction expansion of the Riemann zeta logarithmic derivative, every nearby zero contributes a nonnegative real part at $\sigma +i\gamma$ when $\sigma >1$. Keeping the term from $\rho$ givesAt height zero the pole at one givesand the same local expansion at height $2\gamma$ givesSubstitution yieldsSet $\sigma =1+\eta /\log |\gamma |$, first choosing a sufficiently small absolute $\eta >0$. If $\beta >1-c/\log |\gamma |$, the left side is at least $4\log |\gamma |/(\eta +c)$. Choosing $c>0$ sufficiently small contradicts the last inequality. Therefore