Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 150 / 3 / c / Solution

Shrink the constant $c$ from part b if necessary. Put $L=\log |t|$. Zeros in the disk appearing in the supplied partial-fraction formula have $\log |\gamma |=L+O(1/|t|)$, so part b ensuresfor every such zero.

If $\sigma \ge 1+c/(2L)$, absolute convergence of the logarithmic derivative giveswith the region $\sigma \ge 2$ even easier.

It remains to take $1-c/(2L)<\sigma <1+c/(2L)$. SetFor every local zero, both $\Re (s-\rho )$ and $\Re (s_0-\rho )$ are positive and comparable, while $|s-s_0|\ll 1/L$. The partial-fraction formula at $s_0$, together with the preceding Euler-product bound, givesSince $\Re (s_0-\rho )\gg 1/L$, it follows thatSubtracting the partial-fraction formulas at $s$ and $s_0$ now yieldsTherefore the logarithmic derivative inside the zeta zero-free region satisfiesthroughout the required half-width region.