MATH 7230 - Spring 2013

Analytic Number Theory

This is a first course in elliptic curves and modular forms, which are at the heart of modern number theory.  See the Syllabus and detailed Lecture Schedule for more details.

Course Information

Scheduled Time
Lectures TR 1:30
Lockett 132
Office Hours T 3:00
Lockett 228

Henryk Iwaniec and Emmanuel Kowalski, Analytic Number Theory, American Mathematical
Society, Colloquium Publications 53, 2004

(Optional) G.E. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008.

Problem Sets

Other references


Lecture Schedule

Lecture Topics / Reading / Handouts
Thurs., Jan. 16
Motivating Examples: Twin Prime Conjecture (Y. Zhang; polymath8); Ternary Goldbach Conjecture (Helfgott: arXiv); Proofs of the infinitude of primes (P. Clark)
Tues., Jan. 21
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)
Thurs., Jan. 23
Divisor function bounds; Hyperbola method; Dirichlet divisor problem (Iwaniec-Kowalski 1.5)
Thurs., Jan. 30
Euler summation; Harmonic numbers and Euler-Mascheroni constant; Stirling's Formula; Distribution of primes; Erdos-Turan Conjecture (Iwaniec-Kowalski 4.2)
Tues., Feb. 4
Prime Number Theorem (N. Levinson); Arithmetic functions; Dirichlet convolution; Mobius inversion (Iwaniec-Kowalski 1.3, 1.4, 2.1)
Thurs., Feb. 6
Equivalent statements of PNT (A. Hildebrand); Partial summation; Tchebyshev's bounds (Iwaniec-Kowalski 2.1, 2.2)
Tues., Feb. 11
Levinson's exposition of Selberg's proof (Section 4 of Levinson, A. Selberg); Mertens bound (Iwaniec-Kowalski 2.2, 2.4)
Tues, Feb. 18
Tatuzawa-Iseki's formulation of Selberg's proof (1951); Technical estimates for prime counting functions via partial summation; Selberg's proof of PNT
Thurs., Feb. 20
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
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)
Thurs., Feb. 27
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
Thurs., Mar. 6
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)
Tues., Mar. 11
Stirling's approximation (detailed analysis in S. Dunbar's notes); Asymptotic expansions and divergent series (J. Hunter's notes)
Thurs., Mar. 13
Examples of asymptotic expansions, including the Error function; Borel-Ritt Theorem; Laplace's Method for asymptotic expansions of integrals (M. Evans' notes)
Tues., Mar. 18
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)
Thurs., Mar. 20
Meromorphic continuation of Riemann zeta function; Bernoulli polynomials; Euler-Maclaurin summation (Iwaniec-Kowalski 4.1)
Tues., Mar. 25
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)
Thurs., Mar. 27
Sieve methods; Sieve of Eratosthenes; Legendre's identity (Iwaniec-Kowalski 6.1 - 6.2); Applications to arithmetic density
Tues., Apr. 1
Brun's pure sieve (Iwaniec-Kowalski 6.2); Bonferroni's inequalities (1936); Application to twin primes (Brun 1916)
Thurs., Apr. 3
Brun's general combinatorial sieve (Iwaniec-Kowalski 6.3); Upper and lower bounds
Tues., Apr. 8
Selberg's Upper-bound sieve (Iwaniec-Kowalski 6.5); Cauchy-Schwarz, least squares, and Lagrange multipliers
Thurs., Apr. 10
Application of Selberg's sieve to twin primes, short intervals, and arithmetic progressions (van Lint and Richert 1965)
Tues., Apr. 22
Recent progress with sieves, including prime gaps (Sound's survey; Maynard 2013; Thorner 2014; Castillo-Hall-Lemke Oliver-Pollack-Thompson 2014).
Thurs., Apr. 24
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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