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Prime Number Theorems

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Date: 01/03/99 at 16:19:35
From: Alisa Levine
Subject: Prime Numbers

Hi. I have to write a math paper on prime numbers. I have a few
questions. First of all, what exactly is the prime number theorem? Is
there a way to provide a description in a simpler way to describe it?
I know that it has to do with finding primes and proving primality.
Also, what is the Lucas-Lehmer test for Mersenne primes, and how do
Mersenne primes differ from other primes? What's the Riemann
Hypothesis?

Alisa Levine
```

```
Date: 01/04/99 at 12:09:46
From: Doctor Wilkinson
Subject: Re: Prime Numbers

The Prime Number Theorem provides an approximate answer to the question
"how many prime numbers are there between 1 and n?"  The answer is:

where ln(n) means the natural logarithm of n, which is the power you
need to raise the number e to to get n.  (e is about 2.718281828).

A Mersenne prime is a prime of the form 2^n - 1. You can show by
simple algebra that a number of this form can be prime only if n is
also a prime.

The Lucas-Lehmer test is a way of seeing whether a number of this
form really is prime or not. All the really large primes that are
known are Mersenne primes, because the Lucas-Lehmer test makes it easy
(relatively speaking) to determine whether numbers of this form are
prime or not. Nobody knows whether there is an infinite number of
Mersenne primes. Currently 36 or 37 have been discovered. The test
works as follows:

You let p be a prime and let n = 2^p - 1. Let:

S_1 = 4
S_{k+1} = (S_k)^2 - 2 if k >= 1

Then n is prime if and only if S_{p-1} is divisible by n.

Explaining the Riemann Hypothesis in simple terms is harder, because
This is about as simple as I know how to make it:

The function zeta(s), known as the Riemann Zeta Function, is defined
for complex numbers s for which the real part of s is greater than 1,
by the series:

zeta(s) = sum(1/n^s)   n = 1, ..., infinity

The definition can be extended by a process known as analytic
continuation to all complex values of s, except for s = 1. (We say
that zeta(s) has a simple pole at s = 1). It is fairly easy to prove
that zeta(s) is zero when s is a negative even number. The Riemann
Hypotheses states that all the other zeroes of zeta(s) have imaginary
part equal to 1/2. This is the most famous remaining unsolved problem
in all of mathematics. Its importance for prime numbers is that if it
is true, then we can estimate the number of primes between 1 and n
more accurately.

- Doctor Wilkinson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory

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