eBook - ePub

# Analytic Number Theory for Undergraduates

## Heng Huat Chan

This is a test

Share book

- 128 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android

eBook - ePub

# Analytic Number Theory for Undergraduates

## Heng Huat Chan

Book details

Book preview

Table of contents

Citations

## About This Book

This book is written for undergraduates who wish to learn some basic results in analytic number theory. It covers topics such as Bertrand's Postulate, the Prime Number Theorem and Dirichlet's Theorem of primes in arithmetic progression.

The materials in this book are based on A Hildebrand's 1991 lectures delivered at the University of Illinois at Urbana-Champaign and the author's course conducted at the National University of Singapore from 2001 to 2008.

Contents:

- Facts about Integers
- Arithmetical Functions
- Averages of Arithmetical Functions
- Elementary Results on the Distribution of Primes
- The Prime Number Theorem
- Dirichlet Series
- Primes in Arithmetic Progression

Readership: Final-year undergraduates and first-year graduates with basic knowledge of complex analysis and abstract algebra; academics.

## Frequently asked questions

How do I cancel my subscription?

Can/how do I download books?

At the moment all of our mobile-responsive ePub books are available to download via the app. Most of our PDFs are also available to download and we're working on making the final remaining ones downloadable now. Learn more here.

What is the difference between the pricing plans?

Both plans give you full access to the library and all of Perlego’s features. The only differences are the price and subscription period: With the annual plan you’ll save around 30% compared to 12 months on the monthly plan.

What is Perlego?

We are an online textbook subscription service, where you can get access to an entire online library for less than the price of a single book per month. With over 1 million books across 1000+ topics, we’ve got you covered! Learn more here.

Do you support text-to-speech?

Look out for the read-aloud symbol on your next book to see if you can listen to it. The read-aloud tool reads text aloud for you, highlighting the text as it is being read. You can pause it, speed it up and slow it down. Learn more here.

Is Analytic Number Theory for Undergraduates an online PDF/ePUB?

Yes, you can access Analytic Number Theory for Undergraduates by Heng Huat Chan in PDF and/or ePUB format, as well as other popular books in Mathématiques & Théorie des nombres. We have over one million books available in our catalogue for you to explore.

## Information

Topic

MathématiquesSubtopic

Théorie des nombresChapter 1

The Fundamental Theorem of Arithmetic

1.1 Least Integer Axiom and Mathematical Induction

Let

**Z**= {0, ±1,±2,…}

be the set of integers. The

*Least Integer Axiom*(see [10]), also known as the*Well Ordering Principle*, states that there is a smallest integer in every*nonempty*subset of non-negative integers. It is useful in establishing the following result.**Theorem 1.1.**

*Let S*(1),

*S*(2),…,

*S*(

*n*),…

*be statements, one for each integer n*≥ 1.

*If some of these statements are false, then there is a first false statement.*

*Set*

**Proof.**Since at least one statement is false,

*T*is nonempty. By the Least Integer Axiom, there exists a smallest integer*n*in*T*. This implies that*S*(*n*) is the first false statement.From Theorem 1.1, we deduce the

*Principle of Mathematical Induction.***Theorem 1.2.**

*Let S*(

*n*)

*be statements, one for each*(

*n*≥ 1. Suppose that the following conditions are satisfied by S*n):*

(

*a*)*The statement S*(1)*is true.*(

*b*)*If S*(*n*)*is true, then S*(*n*+ 1)*is true.**Then S*(n)

*is true for all integers n*≥ 1.

*Suppose that*

**Proof.***S*(

*n*) is not true for all integers

*n*≥ 1. Then for some positive integer

*k*≥ 1,

*S*(

*k*) is false. By Theorem 1.1, there is a first false statement, say

*S*(

*m*). By the fact that

*S*(1) is true, we conclude that

*m*≠ 1. Furthermore, by the minimality of

*m*, we observe that

*S*(

*j*) is true for 1 <

*j*≤

*m*– 1. Now, by (b),

*S*(

*m*– 1) is true implies that

*S*(

*m*) is true. This contradicts the assumption that

*S*(

*m*) is false and we conclude that the statements

*S*(

*n*) is true for all positive integers

*n*≥ 1.

We may replace 1 in Theorem 1.2 (a) by any integer

*m*. In other words, we can modify Theorem 1.2 as**Theorem 1.3.**

*Let*(

*m*be an integer. Let S*n*)

*be statements, one for each integer*

*n*≥

*m*.

*Suppose that the following two conditions are satisfied:*

(a)

*The statement**S*(*m*)*is true.*(b)

*If**S*(*n*)*is true, then S*(*n*+ 1)*is true.**Then S*(

*n*)

*is true for all integers n*≥

*m*.

We end this section with another version of the Principle of Mathematical Induction. The proof of this version is similar to the proof of Theorem 1.2 and we leave it as an exercise for the readers.

**Theorem 1.4.**

*Let*(

*m*be an integer. Let S*n*)

*be statements, one for each integer*

*n*≥

*m*.

*Suppose that the following conditions are satisfied:*

(

*a*)*S*(*m*)*is true and*(

*b*)*if S*(*k*)*is true for all m ≤ k ≤*+ 1)*n*then S(n*is true*.*Then S*(

*n*)

*is true for all integers*

*n*≥

*m*.

1.2 Division Algorithm

**Theorem 1.5 (Division Algorithm).**

*Let a and*0.

*b*be integers such that*b*>*Then there exist unique integers*

*q*and*r*with*a*=

*bq*+

*r*,

*where*0 ≤

*r*<

*b*.

**Let**

*Proof.*Note that since

w...