Turing’s method is explained well in p172-175 of the book ‘Riemann’s Zeta Function’ by H.M. Edwards. To understand the finer details, I tried reading the original source of the material in Turing’s paper (https://rauterberg.employee.id.tue.nl/lecturenotes/DDM110%20CAS/Turing/Turing-1953%20Some%20calculations%20on%20the%20Riemann%20zeta-function.pdf), however, I cannot see where in the paper the book gets its explanations from.
The book refers to Gram points $g_n$ and corresponding translations $h_n$ needed to make Gram points $\textit{good}$ Gram points i.e. $(-1)^nZ(g_n+h_n)>0$. It mentions that Turing’s method is about finding sequences $h_m,…, h_{m+k}$ of consecutive $h_n$ values that are sufficiently small and then this is enough to prove $N(g_m)=m+1$.
I do not see where these ideas are written/explained in Turing’s paper. Does the book just use Theorem 4 in the paper and use its own explanation from that point?