Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 23 / 3 / c / Solution

Evaluate the supplied real logarithmic derivative at $2+it$. Since $1<2-\beta <2$, its positive summand is bounded above and below by constant multiples of $(1+|t-\gamma {|}^2{)}^{-1}$. On the other hand, differentiating the defining xi expression givesAt real part two the last term is bounded by the absolutely convergent series $\sum \Lambda (n)n^{-2}$; the gamma logarithmic derivative is $O(\log t)$. ThereforeFor $\gamma \in [T,T+1]$, use $t=T$: each term in the displayed sum is at least $1/2$. Hence the number of zeros in that unit ordinate interval, counted with multiplicities, is $O(\log T)$. These are the local zeta zero-count bound and the corresponding smoothed bound.