Course description

This course is an introduction to applications of probabilistic methods within number theory. We will discuss a selection of the topics presented in [K], starting out from the Erdős-Kac theorem about the distribution of number of distinct prime factors
of a typical integer of size about N . Possible topics include the distribution of values of the Riemann Zeta function, Chebychev bias (which concerns the question whether there are there more primes p ≡ 3 (mod 4) than primes p ≡ 1 (mod 4)) as well as connections between exponential sums and random walks.

Course summary

The main reference is:

[K] E. Kowalski. An Introduction to Probabilistic Number Theory (Cambridge Studies
in Advanced Mathematics). Cambridge University Press, Cambridge, 2021.

Additional references

[B] P. Billingsley, Probability and measure. Anniversary edition Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, 2012.
[E1] P. D. T. A. Elliott, Probabilistic number theory. I. Mean-value theorems. Grundlehren der Mathematischen Wissenschaften 239. Springer-Verlag, New York-Berlin, 1979.
[E2] P. D. T. A. Elliott, Probabilistic number theory. II. Central limit theorems. Grundlehren der Mathematischen Wissenschaften 240. Springer-Verlag, Berlin-New York, 1980.
[G1] A. Gut, Probability: a graduate course. Second edition. Springer Texts in Statistics. Springer, New York, 2013.
[G2] A. Gut, An Intermediate Course in Probability. Second edition. Springer Texts in Statistics.
Springer, New York, 2009.
[H] A. Harper, Probabilistic Number Theory (Michaelmas term 2015), lecture notes.
Available on warwick.ac.uk/fac/sci/maths/people/staff/harper/
[IK] H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004.
[MV ] H.L. Montgomery and R.C. Vaughan. Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics, 97. Cambridge University Press, Cambridge, 2007.
[RS] M. Radziwiłł and K. Soundararajan, Selberg's central limit theorem for $LaTeX: \log |\zeta(\frac{1}{2} + it)|$ . Enseign. Math. 63 (2017), no. 1-2, 1–19. (arXiv version)
[T] G. Tenenbaum, Introduction to analytic and probabilistic number theory. Third edition. Graduate Studies in Mathematics, 163. American Mathematical Society, Providence, RI, 2015.

FSF3634 VT23 Probalistic number theory (61049)