An Introduction to Number Theory
Hor W. H.1 and Pang P. Y. H.2
Department of Mathematics, National University of Singapore
2 Science Drive 2, Singapore 117543
ABSTRACT
The aim of this project is to study the fundamental definitions and certain arithmetical
functions in the theory of numbers. These notions are key ingredients used in proving the
Prime Number Theorem, one of the most important results in number theory. The text
recommended for reading is by Apostol, Tom M. (1976), Introduction to Analytic
Number Theory (Undergraduate Texts in Mathematics), Springer-Verlag. The materials
covered in the project are from chapters 1 and 2, and the relevant exercises in the
chapters are also attempted. The results are then submitted as a report.
INTRODUCTION
In the eighteenth century, Gauss and Legendre independently proposed the Prime
Number Conjecture which states that for large values of x, the number  (x ) of prime
numbers less than or equal to x is asymptotic to x / log x . J. Hadamard and C. J. de la
Vallee Poussin first proved it in 1896. In 1949, Atle Selberg and Paul Erdos gave an
elementary (but not easy) proof using only calculus. The aim of this project is to cover
some of the necessary groundwork that leads to Selberg and Erdos’ proof.
DEFINITIONS
We start with some fundamental concepts in the theory of numbers:
The principle of induction: If Q is a set of integers such that 1  Q and n  Q implies
n + 1  Q, then all integers greater or equal to 1 belong to Q.
The well-ordering principle: If A is a non-empty set of positive integers, then A
contains a smallest number.
An integer p is a prime number if its only positive divisors are 1 and p. If a number is
not prime then it is composite.
The notation d | n denotes that d is a divisor of n.
The greatest common divisor, or g.c.d., of two numbers a and b is denoted by (a, b).
Two numbers a and b are relatively prime if the only common divisor of a and b is 1,
i.e., (a, b) = 1.
Arithmetical functions are real- or complex-valued functions defined on the set of
positive integers, Z+.
Perfect numbers are integers that are equal to the sum of its proper divisors, for
example 6  1  2  3 and 28  1  2  4  7  14 .
1
2
Student
Assistant Professor
1.1 Some arithmetical functions
The Mobius function  (n) is defined such that for k distinct unique primes
 (n) = 1
if n = 1
k
= (1) if n = p1 p2  pk where p1 , p2 , ..., pk are distinct primes
= 0
otherwise.
The Euler totient function  (n ) is defined as the number of positive integers not
exceeding n which are relatively prime to n. It can be expressed as
n
 1 
 ( n)   
.
k 1  n, k  
The Dirichlet product f * g of two arithmetical functions f and g is the arithmetical
function h defined by
n
h(n)   f (d ) g   .
d
d |n
The arithmetical function N is defined by N (n)  n .
The identity function I (n) is given by
1
I (n)    = 1 if n = 1,
n
= 0 if n > 1.
An arithmetical function g is said to be the Dirichlet inverse of f if f * g = g * f = I.
For example, the unit function u(n) = 1 is the Dirichlet inverse of  (n) , i.e.,
I ( n)    ( d ) .
d |n
The Mangoldt function  (n) is defined for every integer n  1 by
 (n)  log p if n = pm for some prime p and some m  1 ,
=0
otherwise.
The Liouville function  (n ) is defined for every integer n  1 such that if
n  p1a1 p2a2  pkak where p1 , p2 , ..., pk are distinct prime factors of n, then
 (n)   1a  a  a .
For a real or complex number  , the divisor function   (n) is defined by
1
2
k
  ( n)   d  .
d |n
In terms of Dirichlet multiplication, we have    u  N  . When  = 0,  0 (n) gives the
number of divisors of n and is usually denoted by d(n). When  = 1,  1 (n) is the sum of
the divisors of n.
1.2 Some properties and constructions involving arithmetical functions
An arithmetical function f is called multiplicative when f is not identically zero and
f (mn)  f (m) f (n)
whenever (m, n) = 1.
It is called completely multiplicative if f (mn)  f (m) f (n) for all m, n.
Let F be a real- or complex-valued function on the domain 0,   with F (x) = 0 for
0  x  1 . For any arithmetical function  , we define the generalized convolution   F ,
such that it is defined on the domain 0,   and vanishes for 0  x  1 , by
  F (n)   (n) F  x  .
n
n x


Given two formal power series A( x)   a (n) x and B( x)   b(n) x n :
n
n 1
n 1
(i) Equality: A( x)  B( x) means that a(n)  b(n) for all n  0 ;

A( x)  B( x)   a(n) b(n) x n ;
(ii) Sum:
n 1

n
n 1
k 0
A( x) B ( x)   c(n) x n where c(n)   a (k )b(n  k ) .
(iii) Product:
For any arithmetical function f and prime p, the Bell series of f modulo p is given by
the formal power series

 
f p ( x)   f p n x n .
n 1
The derivative of any arithmetical function f is given by
f ' (n)  f (n) log n
for n  1 .
The notation f ( x)  Og ( x) (called the Big O notation) means there exists a number
a such that the quotient f ( x) / g ( x) is bounded for x  a , i.e., there exists a constant
M  0 such that
for all x  a .
f ( x)  Mg ( x)
RESULTS AND DISCUSSION
In the text by Apostol quoted, numerous problems are given to aid the thorough
understanding of the above and other concepts. Some of these problems involved the
Mersenne primes, Fibonacci sequence, and perfect numbers. Some results are as follows.
The Mersenne number 2 n  1 being prime implies that n is prime. Any two consecutive
terms in the Fibonacci sequence, defined by the recursion formula an1  an  an1 , are
relatively prime to each other. An integer greater than or equal to 12 is the sum of two
composite numbers. It is also proven that all even perfect numbers can be expressed in
the form 2 a1 2 a  1 where 2 a  1 is a prime. To solve these problems, we derived
alternative expressions for some of the formulas given in the book. For example, the
Mobius inversion formula given in the book can be expressed in both summation and
product form:
n
f ( n )   g ( d )  g ( n)   f ( d )   
d
d |n
d |n
is equivalent to




g ( n)   f ( d ) a ( n / d )

d |n
f ( n)   g ( d ) b ( n / d ) .
d |n
These formulas hold when f (n)  0 for all n, a(n) is a real arithmetical function with
a(1)  0 , and b  a 1 , the Dirichlet inverse of a.
More results as well as details are given in the report.
FURTHER DISCUSSION
The definitions of all the abovementioned arithmetical functions are necessary for the
proof of the Prime Number Theorem which states that
 ( x) log x
lim
1.
x 
x
The required tool for proving the limiting case is to consider the following asymptotic
formula of Selberg:
 x
 ( x) log x    (n)    2 x log x  O( x)
n
n x
where O (x) denotes that the left-hand side is bounded for x  a at some a, and
for x  0 .
 ( x )    ( n)
n x
CONCLUSION
In this project, we studied some of the necessary ingredients in the proof of the Prime
Number Theorem. In particular, we studied in detail properties of prime numbers and
some of the arithmetical functions introduced. The text by Apostol quoted contained
numerous problems in which knowledge of these concepts is tested. We have
successfully solved more than 50 of these problems (in the first two chapters). Details of
some of these solutions are included in the report.
To proceed further with the Prime Number Theorem, further work like finding the
averages of arithmetical functions and Euler’s summation formula remains to be tackled.
REFERENCE
1. Anglin, W.S. (1994) Mathematics: A Concise History and Philosophy
(Undergraduate Texts in Mathematics), Springer-Verlag.
2. Apostol, Tom M. (1976) Introduction to Analytic Number Theory (Undergraduate
Texts in Mathematics), Springer-Verlag.
3. Burton, David M. (1998) Elementary Number Theory, 4th edition, McGraw-Hill
International Editions (Mathematics and Statistics Series).
4. Edited by Cheng-Yih Chen (1987) Science and Technology in Chinese Civilization,
World Scientific.