Summary: For each \(n>2\) we consider the corresponding \(n\)th-partial sum of the Riemann zeta function \(\zeta_n(z):=\sum_{j=1}^nj^{-z}\) and we introduce two real functions \(f_n(c), g_n(c)\), \(c\in \mathbb{R} \), associated with the end-points of the interval of variation of the variable \(x\) of the analytic variety \(| \zeta_n^*(z)| =p_{k_n}^{-c} \), where \(\zeta_n^*(z):=\zeta_n(z)-p_{k_n}^{-z}\) and \(p_{k_n}\) is the last prime not exceeding \(n\). The analysis of fixed point properties of \(f_n, g_n\) and the behavior of such functions allow us to explain the distribution of the real parts of the zeros of \(\zeta_n(z)\). Furthermore, the fixed points of \(f_n, g_n\) characterize the set \(\mathscr{P}^*\) of prime numbers greater than 2 and the set \(\mathscr{C}^*\) of composite numbers greater than 2, proving in this way how close those functions from Arithmetic are. Finally, from the study of the graphs of \(f_n, g_n\) we deduce important properties about the set \(R_{\zeta_n(z)}: =\overline{\{ \mathrm{R}(z):\zeta_n(z)=0\}}\) and the bounds \(a_{\zeta_n(z)}:=\inf \{\mathrm{R}(z):\zeta_n(z)=0\}, b_{\zeta_n(z)}:=\sup \{ \mathrm{R}(z):\zeta_n(z)=0\}\) that define the critical strip \([ a_{\zeta_n(z)},b_{\zeta_n(z)}] \times \mathbb{R}\) where are located all the zeros of \(\zeta_n(z)\).
For the entire collection see [Zbl 1419.46002].