Date
|
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)
|