MATH 7230 - Spring 2018

Analytic Number Theory

This course is an introduction to analytic number theory, the goal of which is to determine precise estimates for arithmetic functions and primes. We will cover classical and modern results, with a particular focus on recent advances in sieve methods (with applications to prime gaps). See the Syllabus and detailed Lecture Schedule for more details.

Course Information

Scheduled Time
Lectures TTh 10:30
Lockett 111
Office Hours T 12:00
Lockett 320

Hugh Montgomery and Robert Vaughan, Multiplicative Number Theory I. Classical Theory,
Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, 2007.
(Available as an e-textbook through LSU libraries)

Problem Sets

Other references

Textbooks and Lecture Notes: 
Papers: (links provided to original sources; subscription or LSU campus-access may be required)

Links to other resources:

Lecture Schedule

Lecture Topics / Reading / Handouts
Thurs., Jan. 12
No class; begin working on HW 1
Tues., Jan. 16
Fundamental Theorem of Arithmetic; Infinitude of primes (P. Clark's lecture notes); Prime Number Theorem; Recent history of prime gaps (Y. Zhang's paper, J. Maynard's paper, Polymath 8 website)
Thurs., Jan. 18
Class canceled due to LSU closure (snow!)
Tues., Jan. 23
Basic properties of divisor function; Highly Composite Numbers (OEIS link); Average value of divisor function (Montgomery-Vaughan [MV] Section 2.1); Discussion of "hyperbola method"
Thurs., Jan. 25
Upper bounds for divisor function, including Wigert and Nicolas-Robin; Discussion of repeated logarithms and "sub-polynomial" functions; Dirichlet's Theorem and the Divisor Problem (Mathworld page); Euler's Summation by Parts (MV Appendix B)
Tues., Jan. 30
Harmonic series and the Euler-Macheroni Constant (MV Section 1.3); Cramér's model for the distribution of primes and the Twin Primes Conjecture (T. Tao's blog); Bounds for n!, Stirling's formula by Summation by parts (MV Appendix B; Section 2.2.3 of A. Hildebrand's notes)
Thurs., Feb. 1
Discussion of Riemann Hypothesis (see A. Granville's survey); Abel's partial summation, and application to Chebyshev's equivalent form of PNT; Logarithmic derivative of Riemann zeta function and von Mangoldt function (MV Section 1.3)
Tues., Feb. 6
Dirichlet series and convolution, including commutative ring structure, Möbius inversion, multiplicative functions
Thurs., Feb. 8
Chebyshev's Theorem via the Hyperbola Method and Stirling's formula (MV Section 2.2); Sums of reciprocal primes (MV Section 2.2)
Thurs., Feb. 15
Continuation of sums of reciprocal primes; Upper bound for divisor function (MV Section 2.3; Nicolas and Robin's paper; T. Tao's blog); Introduction to MathSciNet
Tues., Feb. 20
Convergence properties of Dirichlet series, including abscissae of convergence and absolute convergence, Stolz angles (MV Section 1.2; G. Hardy's book; K. Conrad's notes); Dirichlet's eta-function (MathWorld)
Thurs., Feb. 22
Uniqueness of Dirichlet series on a half-plane and polynomial bounds; Convergence of products of Dirichlet series (MV Section 1.2; F. Bayart's paper); Introduction to the Sieve of Eratosthenes (MV Section 3.1)
Tues., Feb. 27
Sieve of Eratosthenes-Legendre, and failure of "error" term; Bounds for prime counting function; Basic sieve bound for primes (MV Section 3.1)
Thurs., Mar. 1
Selberg's lambda-squared/upper-bound sieve; Applications to upper bound for primes without Chebyshev's Theorem and absolute bound for prime "clustering"; Quadratic optimization and proof of main term (MV Section 3.2)
Tues., Mar. 6
Error term in Selberg's sieve (MV Section 3.2); Further discussion of primes in intervals (H. Maier's paper; D. Hensley and I. Richards' paper)
Thurs., Mar. 8
Selberg's upper-bound Sieve and Twin Primes, main term (MV Section 3.4)
Tues., Mar. 13
Selberg's upper-bound Sieve for general congruence restrictions; error term for Twin Primes (MV Section 3.3, J. Teräväinen's blog)
Thurs., Mar. 15
Introduction to lower bounds from Selberg's sieve and small prime gaps (K. Soundararajan's survey); Admissible k-tuples (Wikipedia page); Hardy-Littlewood Conjecture
Tues., Mar. 20
Introduction to group characters (MV Section 4.2; K. Conrad's blurb); Orthogonality and indicator functions; Multiplicative Dirichlet characters
Thurs., Mar. 22
L-functions of Dirichlet characters (MV Section 4.3, A. Sutherland's notes); Primes in arithmetic progressions; Nonvanishing of L-functions for non-principal characters
Tues., Apr. 3
Examples of Dirichlet characters, including Jacobi symbols; More details of the proof of Dirichlet's Theorem (MV Section 4.3); Prime Number Theorem in Arithmetic Progressions (MV Section 11.3)
Thurs., Apr. 5
Preparation for Zhang and Maynard-Tao's work on small prime gaps, including Dickson tuples; arithmetic/natural density (A. Granville's survey)
Tues., Apr. 10
Von Mangoldt function and Lambda method for controlling number of prime divisors (S. Golomb's paper); Equidistribution of primes in arithmetic progressions, Bombieri-Vinogradov Theorem (A. Granville's survey)
Thurs., Apr. 12
Goldston-Pintz-Yıldırım (GPY) sieve setup, including Selberg weights, combinatorics of "good" residue classes for an admissible k-tuple (A. Granville's survey, Sections 4.1, 4.2)
Tues., Apr. 17
 Diagonalization of quadratic expression via Selberg's construction (A. Granville's survey, Section 4.5); Discussion of generalized divisor functions
Thurs., Apr. 19
Generalized divisor functions and generalization of Hyperbola Method (T. Tao's blog post); Sums of multiplicative functions via Dirichlet Series (A. Granville's survey, Section 4.6), including generalizations of Twin Prime Constant (MV Section 3.4)
Tues., Apr. 21
Discussion of correlations for generalized divisor functions (T. Tao's blog post);  First sum in GPY sieve (A. Granville's survey, Section 4.7) ; Approximation of divisor sums by integrals of smooth functions (A. Granville's survey, Section 4.8)
Thurs., Apr. 26
Higher-dimensional GPY sieve (A. Granville's survey, Section 6.1), and diagonalizing the quadratic form (Section 6.2); Maynard's polynomials and successful use of Bombieri-Vinogradov (J. Maynard's paper)

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