Cramér's conjecture

Abstract: This research paper aims to explicate the complex issue of the Riemann’s Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to demonstrate the maximal prime gaps bound. Maximal prime gaps bound constitutes a comprehensive improvement over the Bertrand’s result, and becomes one of the key elements of the theory. Subsequently, implementing the theory of the primorial function and its error bounds, an improved version of the Gauss’ offset logarithmic integral is developed. The integral serves as the Supremum bound of the prime counting function π(n). Due to its very high precision, it permits to verify the relationship between the prime counting function πn and the offset logarithmic integral of Carl Gauss’. The collective mathematical theory, via the Niels F. Helge von Koch [20] equation: π(n) = Li(n) +O (√ n log(n) ) enables to prove the Riemann’s Hypothesis conclusively. c ⃝2012 Jan Feliksiak 2000 Mathematics Subject Classification. 01A50, 05A10, 0102, 11A41, 11K65, 11L20, 11N05, 1102, 1103.

Abstract: The maximal prime gaps upper bound problem is one of the major mathematical problems to date. The objective of the current research is to develop a standard which will aid in the understanding of the distribution of prime numbers. This paper presents theoretical results which originated with a researchin the subject of the maximal prime gaps. the document presents the sharpest upper bound for the maximal prime gaps ever developed. The result becomes the Supremum bound on the maximal prime gaps and subsequently culminates with the conclusive proof of the Firoozbakht's Hypothesis No 30. Firoozbakht's Hypothesis implies quite a bold conjecture concerning the maximal prime gaps. In fact it imposes one of the strongest maximal prime gaps bounds ever conjectured. Its truth implies the truth of a greater number of known prime gaps conjectures, simultaneously, the Firoozbakht's Hypothesis disproves a known heuristic argument of Granville and Maier. This paper is dedicated to a fellow mathematician, the late Farideh Firoozbakht.

Abstract: I present a new property of prime numbers that leads to a generalization of Cramer's conjecture The study of the gap between consecutive primes is treated as a special case of the gap between consecutive terms of sequences having a certain property called pseudo equidistribution Both rigorous as well as heuristic arguments are given in support of the generalized Cramer's conjectures