A fixed point theory linked to the zeros of the partial sums of the riemann zeta function: In honour of manuel lópez-pellicer

Abstract

For each n>2 we consider the corresponding nth-partial sum of the Riemann zeta function (formula presented) and we introduce two real functions (formula presented), associated with the end-points of the interval of variation of the variable x of the analytic variety (formula presented), where (formula presented) and (formula presented) is the last prime not exceeding n. The analysis of fixed point properties of fn,gn and the behavior of such functions allow us to explain the distribution of the real parts of the zeros of (formula presented). Furthermore, the fixed points of (formula presented) characterize the set (formula presented) of prime numbers greater than 2 and the set (formula presented) 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 (formula presented) we deduce important properties about the set (formula presented) and the bounds (formula presented) that define the critical strip (formula presented) where are located all the zeros of (formula presented).