Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 25 / 3 / a / Solution

Differentiate the locally uniformly convergent logarithm of the Euler product. Its logarithmic derivative isThe differentiated sum is absolutely convergent, since $\Lambda (n)\le \log n$. Taking the real parts at the three heights givesEach summand is nonnegative because its bracket is $2(1+\cos (t\log n){)}^2$. This proves the derivative form of the three-four-one zero-free-region argument.