*“Using randomness to study certainty may seem somewhat surprising.*

It is, however, one of the deepest contributions of our century to mathematics in general and to the theory of numbers in particular.”

It is, however, one of the deepest contributions of our century to mathematics in general and to the theory of numbers in particular.”

Gérald Tenenbaum, Michel Mendès France,

*The Prime Numbers and Their Distribution*. AMS, 2000

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Here is the list of questions and problems of Number Theory listed below I tried to address entirely from the probabilistic point of view. I am hoping that some of these questions were answered satisfactory in these notes, while others are still unanswered waiting for more detailed analysis and resolution.

among arithmetic sequences, are addressed here from an entirely probabilistic point of view. We discuss the concepts of ‘randomness’ and ‘independence’ relevant to number-theoretic problems and interpret the basic concepts of divisibility of natural number in terms of probability spaces and appropriate probability distributions on classes of congruence. We analyze and demonstrate the importance of Zeta probability distribution to prove, in particular, theorems stating the equivalence of probabilistic independence of divisibility by co-prime factors, and the fact that random variables with the property of independence of co-prime factors must have Zeta probability distribution.

Multiplicative and additive models with recurrent equations for generating sequences of prime numbers are derived based on the reduced Sieve of Eratosthenes Algorithm. This allows to interpret such sequences as realizations of random walks on set of natural numbers and on multiplicative semigroups generated by set of odd prime numbers , representing paths of stochastic dynamical systems.

The Cramér's model for probability distribution of primes is modified and applied to analyze the sequences of primes generated by appropriate random walks. This leads to study of prime number distributions among arithmetic sequences (classes of congruence) based on expectations and variances for occurrences of primes not exceeding in the arithmetic sequences. We illustrate computer calculations supporting the conjecture of the uniform distribution of primes among congruence classes for each given prime. By using Zeta probability distribution and the modified Cramér’s model we approach a number of Number Theory problems, including the Strong Goldbach Conjecture (SGC). The solution to the Twin Primes problem and, more general, prime distribution problem for consecutive d-prime

numbers is also suggested in this paper.

**Abstract**Here is the list of questions and problems of Number Theory listed below I tried to address entirely from the probabilistic point of view. I am hoping that some of these questions were answered satisfactory in these notes, while others are still unanswered waiting for more detailed analysis and resolution.

- Additive and sigma-additive probability measures on the set of natural numbers
- Probability distributions on multiplicative semigroups of natural numbers and Riemann Zeta function.
- Cramér's model and Zeta distribution of prime numbers
- Approximations of the counting function of prime numbers Pi(x)
- Additive and multiplicative models of random walks on a sets of prime numbers.
- Probability modeling for residuals and the Chinese Reminder Theorem

among arithmetic sequences, are addressed here from an entirely probabilistic point of view. We discuss the concepts of ‘randomness’ and ‘independence’ relevant to number-theoretic problems and interpret the basic concepts of divisibility of natural number in terms of probability spaces and appropriate probability distributions on classes of congruence. We analyze and demonstrate the importance of Zeta probability distribution to prove, in particular, theorems stating the equivalence of probabilistic independence of divisibility by co-prime factors, and the fact that random variables with the property of independence of co-prime factors must have Zeta probability distribution.

Multiplicative and additive models with recurrent equations for generating sequences of prime numbers are derived based on the reduced Sieve of Eratosthenes Algorithm. This allows to interpret such sequences as realizations of random walks on set of natural numbers and on multiplicative semigroups generated by set of odd prime numbers , representing paths of stochastic dynamical systems.

The Cramér's model for probability distribution of primes is modified and applied to analyze the sequences of primes generated by appropriate random walks. This leads to study of prime number distributions among arithmetic sequences (classes of congruence) based on expectations and variances for occurrences of primes not exceeding in the arithmetic sequences. We illustrate computer calculations supporting the conjecture of the uniform distribution of primes among congruence classes for each given prime. By using Zeta probability distribution and the modified Cramér’s model we approach a number of Number Theory problems, including the Strong Goldbach Conjecture (SGC). The solution to the Twin Primes problem and, more general, prime distribution problem for consecutive d-prime

numbers is also suggested in this paper.