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Analytic Number Theory (MATH M0007)

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Administrative Information

  1. Unit number and title: MATH M0007 Analytic Number Theory
  2. Level: M/7
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 2009
  6. Unit Organiser: Alan Haynes
  7. Lecturer: Alan Haynes
  8. Teaching block: 1
  9. Prerequisites: Complex Function Theory (Math 33000), Number Theory and Group Theory (Math 11511)

Unit aims

To gain an understanding and appreciation of analytic number theory, and some of its most important achievements. To be able to apply the techniques of complex analysis to study a range of specific problems in number theory.

General Description of the Unit

The study of prime numbers is one of the most ancient and beautiful topics in mathematics. After reviewing some basic results in elementary number theory and the theory of Dirichlet characters and L-functions, the main aim of this lecture course will be to show how the power of complex analysis can be used to shed light on irregularities in the sequence of primes. Significant attention will be paid to developing the theory of the Riemann zeta function. The course will build up to a proof of the Prime Number Theorem and a description of the Riemann Hypothesis, arguably the most important unsolved problem in modern mathematics.

Relation to Other Units

This is one of three Level 6 and Level 7 units which develop number theory in various directions. The others are Number Theory and Algebraic Number Theory.

Teaching Methods

Lectures and exercises.

Learning Objectives

To gain an understanding and appreciation of Analytic Number Theory and some of its important applications. To be able to use the theory in specific examples.

Assessment Methods

The assessment mark for Analytic Number Theory is calculated from a 2½-hour written examination in April consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted in this examination.

Award of Credit Points

To gain credit points for this unit, students must gain a pass mark (50 or over) for the unit

Transferable Skills

Using an abstract framework to better understand how to attack a concrete problem.

Texts

  1. T. M. Apostol, Introduction to analytic number theory. Springer, 1976.
  2. J. Brüdern, Einführung in die analytische Zahlentheorie, Springer, 1995.
  3. H. Davenport, Multiplicative Number Theory, third edition, Springer 2000
  4. H.L. Montgomery and R.C. Vaughan, Multiplicative Number Theory. I. Classical Theory, Cambridge University Press 2007
  5. G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995.

Syllabus

Preliminary review. Elementary results on the distribution of primes. Equivalent forms of the Prime Number Theorem. Chebychev's order of magnitude result for the prime counting function.

Definition and basic properties of arithmetic functions. Dirichlet convolution and the ring of arithmetic functions. Mobius inversion.

Summation techniques and average orders of arithmetic functions. Partial summation. Discussion of the Dirichlet divisor problem.

Dirichlet series and Euler products.

Dirichlet characters and Dirichlet L-functions. Dirichlet's theorem on primes in arithmetic progression.

Analytic properties of the Riemann zeta function. The gamma function and the functional equation for the zeta function. The Riemann Hypothesis.

Size of the zeta function in the critical strip. Non-vanishing of the zeta function on the line s=1. Perron's formula. Analytic proof of the prime number theorem. Equivalent forms of the Riemann Hypothesis.

Other possible topics as time permits: More about primes in arithmetic progressions, sieve techniques, additive number theory.