@ARTICLE{10.21494/ISTE.OP.2024.1159, 

TITLE={The Riemann zeta function or the umbilicus of mathematics (II): about the foundations}, 
  
AUTHOR={Philippe Riot, }, 
	
JOURNAL={Entropy: Thermodynamics – Energy – Environment – Economy }, 
	
VOLUME={5}, 

NUMBER={Special issue LILA 2}, 
	
YEAR={2024}, 

URL={https://openscience.fr/The-Riemann-zeta-function-or-the-umbilicus-of-mathematics-II-about-the}, 
	
DOI={10.21494/ISTE.OP.2024.1159}, 
	
ISSN={2634-1476}, 
	
ABSTRACT={Whereas the part 1 sentence attributed to Heraclitus « εν και παν » is a whole merit of being revisited. The significance of the Riemann zeta function ζ emerges using an ordinal reading of this function. In this context, it is important to complete the basic axioms attached to ZFC set theory (Zermelo-Fraenckel with axiom of choice) in order to precisely characterize the cardinality of the continuum. The very definition of the function justifies retaining Martin’s axiom (allowing the rules of infinite combinatorics to be extended beyond the threshold of the smallest uncountable ordinal), the simplest forcing axiom which avoids any argument from metamathematical nature. It then turns out that this is reformulated in an equivalent way like Riemann’s statement on the distribution of its non-trivial zeros.}}