Date
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Lecture Topics / Reading / Handouts
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Thurs., Jan. 16
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Motivating
Examples: Twin Prime Conjecture (Y. Zhang; polymath8);
Ternary Goldbach Conjecture (Helfgott: arXiv); Proofs
of the infinitude of primes (P. Clark)
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Tues., Jan. 21
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Proofs of
the infinite of the primes (cont.); Riemann Zeta-function;
Transcendence and Lindemann-Weierstrass theorem (Proofs
of irrationality of pi); Divisor function (Iwaniec-Kowalski
1.2)
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Thurs., Jan. 23
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Divisor
function bounds; Hyperbola method; Dirichlet divisor problem
(Iwaniec-Kowalski 1.5) |
Thurs., Jan. 30
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Euler
summation; Harmonic numbers and Euler-Mascheroni constant;
Stirling's Formula; Distribution of primes; Erdos-Turan
Conjecture (Iwaniec-Kowalski 4.2) |
Tues., Feb. 4
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Prime
Number Theorem (N.
Levinson); Arithmetic functions; Dirichlet
convolution; Mobius inversion (Iwaniec-Kowalski 1.3, 1.4,
2.1) |
Thurs., Feb. 6
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Equivalent statements of PNT
(A.
Hildebrand); Partial summation; Tchebyshev's bounds
(Iwaniec-Kowalski 2.1, 2.2) |
Tues., Feb. 11
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Levinson's
exposition of Selberg's proof (Section 4 of Levinson, A.
Selberg); Mertens bound (Iwaniec-Kowalski 2.2, 2.4) |
Tues, Feb. 18
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Tatuzawa-Iseki's
formulation of Selberg's proof (1951);
Technical estimates for prime counting functions via partial
summation; Selberg's proof of PNT
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Thurs., Feb. 20
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Dirichlet's
Theorem; Characters of finite abelian groups; Orthogonality,
Fourier analysis, and indicator functions (Iwaniec-Kowalski
2.3, 3.1. For additional reference, see K. Conrad's notes) |
Tues., Feb. 25
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Additive
and multiplicative Dirichlet Characters (Iwaniec-Kowalski
2.3, 3.2); Dirichlet series associated to characters; Simple
bounds for L-functions of non-principal characters (See 17.1
- 17.2 of P. Clark's notes)
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Thurs., Feb. 27
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Meromorphic
continuation of Riemann zeta function; Nonvanishing of
L(1,\chi) for nonprincipal characters (17.3 of P. Clark's notes;
Alternative proof in A. Hildebrandt's notes);
Proof of Dirichlet's theorem; Introduction to arithmetic
density
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Thurs., Mar. 6
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Arithmetic
density vs. logarithmic (Dirichlet) density; Hadamard-de la
Vallee Poussin's theorem and arithmetic density of primes in
arithmetic progressions (Summary in Secton 4 of H. Kun's paper);
Definition and basic properties of Gamma function (M. Evans'
notes)
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Tues., Mar. 11
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Stirling's
approximation (detailed analysis in S. Dunbar's notes);
Asymptotic expansions and divergent series (J. Hunter's notes)
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Thurs., Mar. 13
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Examples
of asymptotic expansions, including the Error function;
Borel-Ritt Theorem; Laplace's Method for asymptotic
expansions of integrals (M. Evans' notes)
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Tues., Mar. 18
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Fourier
series as generating functions; Theta function; the Mellin
transform (J.-P. Ovarlez's textbook),
and asymptotic expansions (D. Zagier's appendix);
Bernoulli numbers (P. Guerzhoy's notes)
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Thurs., Mar. 20
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Meromorphic
continuation of Riemann zeta function; Bernoulli
polynomials; Euler-Maclaurin summation (Iwaniec-Kowalski
4.1)
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Tues., Mar. 25
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Asymptotic
expansions using Euler-Maclaurin summation (D. Zagier's appendix);
Examples of asymptotic expansions, including theta
functions; Applications to partitions (see the appendix of
my paper)
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Thurs., Mar. 27
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Sieve
methods; Sieve of Eratosthenes; Legendre's identity
(Iwaniec-Kowalski 6.1 - 6.2); Applications to arithmetic
density
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Tues., Apr. 1
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Brun's
pure sieve (Iwaniec-Kowalski 6.2); Bonferroni's inequalities
(1936);
Application to twin primes (Brun 1916)
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Thurs., Apr. 3
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Brun's
general combinatorial sieve (Iwaniec-Kowalski 6.3); Upper
and lower bounds
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Tues., Apr. 8
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Selberg's
Upper-bound sieve (Iwaniec-Kowalski 6.5); Cauchy-Schwarz,
least squares, and Lagrange multipliers |
Thurs., Apr. 10
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Application
of Selberg's sieve to twin primes, short intervals, and
arithmetic progressions (van Lint and Richert 1965)
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Tues., Apr. 22
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Recent
progress with sieves, including prime gaps (Sound's survey;
Maynard 2013;
Thorner 2014;
Castillo-Hall-Lemke Oliver-Pollack-Thompson 2014).
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Thurs., Apr. 24
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Complex
analysis and Hadamard and de la Vallee Poussin's proof of
PNT (Section 3 of Granville's survey);
Mellin-Perron formula (MathOverFlow discussion);
Riemann Hypothesis (Clay Millennium Problem)
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