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

The truncated Perron formula says that, for $c>1$ and $T\ge 2$,where an endpoint has half weight and the error is controlled by

Apply this with $a_n=\mu (n)$, $c=1+1/\log x$, andPart (b) makes the integrand $x^s/(s\zeta (s))$. Move the contour tousing a contour that stays inside the Zero-free region of the Riemann zeta function near small $|t|$. The estimates supplied in the question give $1/\zeta (s)\ll \log (|t|+3)$ on the new contour. Its vertical segment is thereforeafter decreasing $c_3$. The horizontal segments and the Perron truncation error areNo residue is crossed because $1/\zeta (s)$ has a zero, rather than a pole, at $s=1$. Thus the Mertens function satisfiesfor some absolute $c>0$.