MAT 608: Topics in analytic number theory

Fall 2018

Robert Hough

Assistant Professor, Mathematics
SUNY Stony Brook

Send the lecturer (R. Hough) email at: robert.hough - at -

Office: 4-118 Mathematics Building

Office hours: 6-7pm Monday in the MLC, 9-11 am, Friday in 4-118.

Lectures: MW 10:00-11:20pm Physics P122

This course combines an introduction to analytic number theory with a selection of modern topics in prime and combinatorial number theory. Roughly the first third of the course will cover the Prime Number Theorem in arithmetic progressions, following the treatment of Davenport's Multiplicative number theory. The remaining topics are to include the proof of Linnik's Theorem on the least prime in an arithmetic progression and the Bombieri-Vinogradov Theorem, following the treatment of Bombieri's The large sieve in analytic number theory, Maynard's proof of bounded gaps between primes, and the Green-Tao Theorem on arbitrarily long arithmetic progressions in the primes.


Tentative Lecture Schedule

Mon 8/271. Davenport Chap. 1 Primes in arithmetic progressions
Wed 8/292. Davenport Chap. 2-3 Gauss sums, cyclotomy
Mon 9/3 No class Labor day
Wed 9/53. Davenport Chap. 4-5 Primes in arithmetic progressions, primitive characters
Mon 9/104. Davenport Chap. 6 Dirichlet's Class Number Formula
Wed 9/125. Davenport Chap. 7-9 The distribution of primes, Riemann's memoir, the functional equation
Mon 9/176. Davenport Chap. 10-12Properties of the Gamma function, integral functions of order 1, the infinite product representations
Wed 9/197. Davenport Chap. 13-14 The classical zero free region
Mon 9/248. Davenport Chap. 15-17 The zero counting functions, and the explicit formula
Wed 9/269. Davenport Chap. 18-20 The prime number theorem, PNT in AP I
Mon 10/110. Davenport Chap. 21-22Siegel's theorem, PNT in AP II
Wed 10/311. Bombieri Chap. 0-2The analytic large sieve
Mon 10/8 No classFall break
Wed 10/1012. Bombieri Chap. 3-4 Selberg's sieve, the multiplicative large sieve
Mon 10/1513. Montgomery Chap. 5 Turan's method
Wed 10/1714.Bombieri Chap. 6Linnik's theorem
Mon 10/2215.Bombieri Chap. 7Bombieri-Vinogradov Theorem
Wed 10/2416. Maynard Sec. 4The Goldston-Pintz-Yilidirim method
Mon 10/2917. Maynard Sec. 5-6Exposing the main terms
Wed 10/3118. Maynard Sec. 7 The optimization
Mon 11/519. Roth's theorem
Wed 11/720. Gowers sec. 3The uniformity norms
Mon 11/1221. Green-Tao sec. 3 Pseudo-random measures
Wed 11/1422. Green-Tao sec. 5 The generalized von Neumann inequality
Mon 11/1923. Green-Tao sec. 6 The dual norms
Wed 11/21 No class Thanksgiving
Mon 11/2624. Green-Tao sec. 7-8 Koopman-von Neumann Theorem
Wed 11/2825. Green-Tao sec. 8 Proof of the relative Szemerédi Theorem
Mon 12/326. Green-Tao sec. 9 Construction of the pseudo-random measure
Wed 12/527. The Lovász Local Lemma
Mon 12/1028. Hough Covering systems of congruences

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