Combine parts (a)--(c). For every $k\geq1$,
$$
\tau_p(u)^k
\leq\tau_p(ku)
\leq(\mathbb E_pM_m)^{\lfloor k|u|/m\rfloor}
\leq r^{m\lfloor k|u|/m\rfloor}.
$$
Taking $k$th roots and letting $k\to\infty$ gives
$$
\mathbb P_p(0\longleftrightarrow u)=\tau_p(u)
\leq r^{|u|}
=\left(1-\chi(p)^{-1}\right)^{|u|}.
$$

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 212 / 4 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-212/4/d/solution