Introduction to Analytic Number Theory Math 531 Lecture Notes …
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Introduction to Analytic Number Theory. Math 531 Lecture Notes, Fall 2005. A.J. Hildebrand. Department of Mathematics. University of Illinois …
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Introduction to Analytic Number Theory
Math 531 Lecture Notes, Fall 2005
A.J. Hildebrand
Department of Mathematics
University of Illinois
http://www.math.uiuc.edu/~hildebr/ant
Version 2006.09.01Chapter 0
Primes and the Fundamental
Theorem of Arithmetic
Primes constitute the holy grail of analytic number theory, and many of
the famous theorems and problems in number theory are statements about
primes. Analytic number theory provides some powerful tools to study prime
numbers, and most of our current (still rather limited) knowledge of primes
has been obtained using these tools.
In this chapter, we give a precise de nition of the concept of a prime,
and we state the Fundamental Theorem of Arithmetic, which says that every
integer greater than 1 has a unique (up to order) representation as a product
of primes. We conclude the chapter by proving the in nitude of primes.
The material presented in this chapter belongs to elementary (rather
than analytic) number theory, but we include it here in order to make the
course as self-contained as possible.
0.1 Divisibility and primes
In order to de ne the concept of a prime, we rst need to de ne the notion
of divisibility.
Given two integers d 6= 0 and n, we say that d divides n or n is
divisible by d, if there exists…
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