MATHSCRIPTUM
  • Home
  • Mathematics
    • Number Theory
    • Random Walks on Semigroups Generated by Primes
    • The Riemann Zeta Function and Sums of Independent Random Variables with Geometric Distribution
    • Dynamical Systems
    • Probability and Statistics
    • Stochastic Processes
    • Random Fields
    • Lomonosov Moscow State University
    • Applications >
      • Math Class
      • Physics
      • Biology
      • Sociology
    • Иноходец. Урок Перельмана.
  • Poetry
    • Life and Love >
      • To My Youngest Daughter on her 18th Birthday
      • Давай договоримся... Let's negotiate...
      • To Anna
      • Мы рядышком с тобою были...
      • Весна Осенью
      • I called you three times...
      • Я подозрителен и болен…
      • Ушла жена…
    • Spiritual >
      • Баллада о Богатыре Изе
      • Я не здешний...
      • Jerusalem
      • My chariot
      • The day my rent would come to end...
      • Questions to the Almighty G-d
      • В деньгах ли счастье?
    • Science and Math >
      • Прыжок в другое измеренье
      • Божественные числа. Divine Numbers
      • Неужто мир наш автомат…
      • Thunderstorm
    • Translations >
      • A Psalm of Life
      • When I'm 64 - The Beatles
    • Literary >
      • Alliteration
      • Изящная словесность
      • Поединок поэтов в Сети
  • Blog
  • Всё о нас
  • Random Walks on Semigroups of Primes
  • Random Walks on Semigroups of Primes
  • Solving Goldbach Conjecture
  • New Page
  • New Page

 A Probabilistic Approach
to some Problems of Number Theory

​“…Mathematics is the art of giving the same name to different things…The only facts worthy of our attention are those which introduce order into this complexity and so make it accessible to us”.
​
(Henry Poincaré, The Value of Science, Random House, Inc., 2001.

“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.”

                          Gérald Tenenbaum, Michel Mendès France, The Prime Numbers and Their Distribution. AMS, 2000
_a_prob_appr_to__add_and_mult_problems_of_nt.docx
File Size: 0 kb
File Type: docx
Download File

​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.
  1. Additive and  sigma-additive probability measures on the set of natural numbers
  2. Probability distributions on multiplicative semigroups of natural numbers and Riemann Zeta function.
  3. Cramér's model and Zeta distribution of prime numbers
  4. Approximations of  the counting function of prime numbers Pi(x)
  5. Additive and multiplicative models of random walks on a sets of prime numbers.
  6. Probability modeling for residuals and the Chinese Reminder Theorem
Some classical questions and problems of Number Theory, like the Goldbach conjecture, distributions of twin- and primes and primes
​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.