Math 587: Introduction to Analytic Number Theory.

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Catalog Information

Title

Introduction to Analytic Number Theory.

Credit Hours

(3:3:0)

Offered

Contact Department

Prerequisite

(Math 352) or equivalent; instructor's consent.

Description

Arithmetical functions; distribution of primes; Dirichlet characters; Dirichlet's theorem; Gauss sums; primitive roots; Dirichlet L-functions; Riemann zeta-function; prime number theorem; partitions.

Desired Learning Outcomes

Students should gain a familiarity with the problems and tools of analytic number theory at beginning graduate level.

Prerequisites

A knowledge of complex analysis at the level of a first course such as Math 352 should suffice.

Minimal learning outcomes

Students should be familiar with the following concepts. They should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving simple results.

  1. Arithmetic functions
    • convolution of arithmetic functions
    • Möbius inversion
    • multiplication function
    • Euler's summation formula, Abel's identity
    • average order of τ(n), σ(n) and φ(n)
  2. Elementary theorems on distribution of prime numbers
    • Chebyshev's inequalities
    • Mertens' theorem
  3. Finite abelian groups and their characters
    • characters of finite abelian groups
    • the character group
    • Dirichlet characters
    • nonvanishing of L(1, χ) for real nonprincipal χ
    • Gauss sums associated with Dirichlet characters
    • Polya's inequality
    • Dirichlet's theorem on primes in arithmetic progression
  4. Dirichlet series and Euler product
    • Half plane of absolute convergence
    • multiplication of Dirichlet series
    • Euler products
    • Perron's formula for partial sums
  5. The zeta function and the Dirichlet L-functions
    • Properties of the gamma function
    • Hurwitz zeta function
    • Analytic continuation of ζ(s) and L(s, χ)
    • functional equation for ζ(s) and L(s, χ)
    • nonvanishing of ζ(s) on σ = 1
  6. Partitions
    • Generating functions for partitions
    • Euler's pentagonal-number theorem
    • Jacobi's triple product identity and its consequences.


Textbooks

Possible textbooks for this course include (but are not limited to):

Additional topics

These are at the discretion of the instructor as time allows.

Courses for which this course is prerequisite

None