# Difference between revisions of "Math 587: Introduction to Analytic Number Theory."

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#*functional equation for ζ(''s'') and ''L''(''s'', χ) | #*functional equation for ζ(''s'') and ''L''(''s'', χ) | ||

#*nonvanishing of ζ(''s'') on σ = 1 | #*nonvanishing of ζ(''s'') on σ = 1 | ||

+ | #'''Partitions''' | ||

+ | #*Generating functions for partitions | ||

+ | #*Euler's pentagonal-number theorem | ||

+ | #*Jacobi's triple product identity and its consequences. | ||

</ul> | </ul> | ||

## Revision as of 09:45, 25 May 2010

## Contents

## Catalog Information

### Title

Introduction to Analytic Number Theory.

### Credit Hours

(3:3:0)

### Offered

F

### 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.

**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*)

**Elementary theorems on distribution of prime numbers**- Chebyshev's inequalities
- Asymptotic formula for :<math>\sum_{p\le x} 1/p</math>

**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

**Dirichlet series and Euler product**- Half plane of absolute convergence
- multiplication of Dirichlet series
- Euler products
- Perron's formula for partial sums

**The functions ζ(***s*) and*L*(*s*, &chi';)- Properties of the gamma function
- Harwitz 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

**Partitions**- Generating functions for partitions
- Euler's pentagonal-number theorem
- Jacobi's triple product identity and its consequences.