Probabilistic Methods in Number Theory: Recent Developments
Main Article Content
Abstract
Probabilistic methods have revolutionized the study of number theory, offering powerful tools for understanding the distribution of prime numbers and other fundamental properties of integers. This paper provides an overview of probabilistic methods in number theory, focusing on recent developments and applications. We begin with a discussion of the basic concepts in number theory, including prime numbers, divisibility, and congruences. We then explore the history and background of probabilistic methods, highlighting key developments such as the probabilistic prime number theorem and random matrix theory. Next, we discuss applications of probabilistic methods in prime number distribution and the use of random matrix models in number theory. We also examine recent advances in probabilistic number theory, including new results and conjectures. Finally, we discuss the challenges and future directions of probabilistic number theory, including computational challenges, bridging the gap between theory and practice, and the potential for further innovation.
Downloads
Metrics
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
You are free to:
- Share — copy and redistribute the material in any medium or format for any purpose, even commercially.
- Adapt — remix, transform, and build upon the material for any purpose, even commercially.
- The licensor cannot revoke these freedoms as long as you follow the license terms.
Under the following terms:
- Attribution — You must give appropriate credit , provide a link to the license, and indicate if changes were made . You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
Notices:
You do not have to comply with the license for elements of the material in the public domain or where your use is permitted by an applicable exception or limitation .
No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material.
References
Zhang, Yitang. "Bounded gaps between primes." Annals of Mathematics 179.3 (2014): 1121-1174.
Maynard, James. "Small gaps between primes." Annals of Mathematics 181.1 (2015): 383-413.
Tao, Terence. "The Gaussian Unitary Ensemble and the distribution of primes." Blog post, 2007.
Conrey, John B., et al. "Integral moments of L-functions." Proceedings of the London Mathematical Society
3 (2005): 33-104.
Goldston, D. A., et al. "Small gaps between primes or almost primes." Transactions of the American
Mathematical Society 361.9 (2009): 5285-5330.
Granville, Andrew. "Harald Cramér and the distribution of prime numbers." Scandinavian Actuarial Journal
(1995): 12-28.
Heath-Brown, D. R. "Almost-prime k-tuples." Journal of the London Mathematical Society 2.3 (1978):
-238.
Hardy, G. H., and J. E. Littlewood. "Some problems of ‘partitio numerorum’; III: On the expression of a
number as a sum of primes." Acta Mathematica 44.1 (1923): 1-70.
Selberg, Atle. "An elementary proof of the prime-number theorem." Annals of Mathematics 50.2 (1949):
-313.
Montgomery, Hugh L. "The pair correlation of zeros of the zeta function." Analytic number theory.
Springer, Berlin, Heidelberg, 1974. 181-193.
Erdős, Paul. "On a lemma of Littlewood and Offord." Bulletin of the American Mathematical Society 51.12
(1945): 898-902.
Hardy, G. H., and E. M. Wright. An introduction to the theory of numbers. Oxford University Press, USA,
Tao, Terence, and Van Vu. Additive Combinatorics. Cambridge University Press, 2006.
Nathanson, Melvyn B. Additive Number Theory: The Classical Bases. Springer Science & Business Media,
Riemann, Bernhard. "On the number of primes less than a given magnitude." Translated from the German
by A. Fröhlich, in Collected Papers of Bernhard Riemann (1862).
Rankin, Robert A. "The difference between consecutive prime numbers." Journal of the London
Mathematical Society 13.4 (1938): 242-247.
Cramér, Harald. "On the order of magnitude of the difference between consecutive prime numbers." Acta
Arithmetica 2.1 (1936): 23-46.
Hardy, G. H., and E. M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, 2008.
Hardy, G. H., and J. E. Littlewood. "Some problems of ‘partitio numerorum’; III: On the expression of a
number as a sum of primes." Acta Mathematica 44.1 (1923): 1-70.
Riemann, Bernhard. "On the number of primes less than a given magnitude." Translated from the German
by A. Fröhlich, in Collected Papers of Bernhard Riemann (1862).
Rankin, Robert A. "The difference between consecutive prime numbers." Journal of the London
Mathematical Society 13.4 (1938): 242-247.
Cramér, Harald. "On the order of magnitude of the difference between consecutive prime numbers." Acta
Arithmetica 2.1 (1936): 23-46.
Iwaniec, Henryk, and John Friedlander. Opera de cribro. Vol. 57. American Mathematical Soc., 2010.
Goldfeld, Dorian, and Joseph Hundley. Automorphic representations and L-functions for the general linear
group. Vol. 53. Cambridge University Press, 2011.
Soundararajan, Kannan. "Moments of the Riemann zeta-function." The Riemann zeta-function. Springer,
Berlin, Heidelberg, 2001. 1-36.
Conrey, John B. "More than two fifths of the zeros of the Riemann zeta function are on the critical line."
Journal of the American Mathematical Society 19.3 (2006): 797-803.
Hardy, G. H., and E. M. Wright. An introduction to the theory of numbers. Oxford University Press, USA,
Montgomery, Hugh L. "The pair correlation of zeros of the zeta function." Analytic number theory.
Springer, Berlin, Heidelberg, 1974. 181-193.
Tao, Terence, and Van Vu. Additive Combinatorics. Cambridge University Press, 2006.
Nathanson, Melvyn B. Additive Number Theory: The Classical Bases. Springer Science & Business Media,