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

Put $L=\log t$ and $\ell =\log L$, for sufficiently large $t$. Apply the Landau zero-free-region theorem withwhere $a>0$ is fixed and small enough that $\eta \le 1/4$. On its two discs, the real part is at least $1-3\eta /4$ and the imaginary part is comparable to $t$. The given Richert bound for the Riemann zeta function therefore gives, on the part left of one,On the part right of one, the separately given $O({\log }^{2/3}t)$ bound gives the same conclusion. We may thus choose $M=L^A$ for one fixed sufficiently large $A$. Also $\log (1/\eta )=O(\ell )$, so the logarithmic term in the Landau zero-free-region theorem is $O(\ell )$. Its conclusion isfor large $t$ and a sufficiently small positive $c$. Complex conjugation supplies negative heights. This is the Vinogradov-Korobov zero-free region. Only the stated Richert upper bounds, the Euler product, the pole at one and the proved Landau zero-free-region theorem were used; no prior zero-free-region theorem was assumed.