Abstract
This work is divided into two parts. In the first one, the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed, and new integral lower and upper bounds of π(x) are found.
Highlights
This work aims to investigate two main objectives: what random means in the case of the distribution of prime numbers and the general conditions ensuring that what we know as Prime Number Theorem and Riemann Hypothesis hold in the case of a generic random sequence of natural numbers
The first aim of this work was to deepen the problem of randomness in the distribution of prime numbers through such simple combinatorial objects as First Occurrence Sequences, showing new analogies between the classical set partition problem and the distribution of primes themselves
First Occurrence Sequences define a general class of objects for which the Prime Number Theorem holds, such as for the prime sequence, but they fail to represent more stringent constraints, required by the Riemann Hypothesis, such as the equivalent condition established by Helge von Koch I called RH rule
Summary
This work aims to investigate two main objectives: what random means in the case of the distribution of prime numbers and the general conditions ensuring that what we know as Prime Number Theorem and Riemann Hypothesis hold in the case of a generic random sequence of natural numbers To achieve these goals, we will make use of models based on combinatorics and probability theory. The following sections are devoted to developing the theory of FOS and their analogies with the distribution of primes Despite their simplicity, FOS may serve as a general model of random integer sequences which can be applied in different contexts, from combinatorics to physical models (I give a brief hint about this in the last section), showing the common features to all these fields, such as what is known as PNT.
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