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### The Riemann Zeta Function

```
Date: 10/11/98 at 13:14:56
From: David Bandel
Subject: Unsolved Problems

What is the Riemann Hypothesis?
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```
Date: 10/11/98 at 15:29:36
From: Doctor Tom
Subject: Re: Unsolved Problems

Hi David,

It's a bit tough to explain unless you know quite a bit of
mathematics. The Riemann zeta function is defined as follows:

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

or, if you like:

zeta(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + ...

This makes perfect sense for real numbers s > 1, but there is a
differentiable complex function that exactly agrees with the zeta
described above on the reals greater than 1, and which makes sense over
the entire complex plane, except at s = 1.

This function has an infinite number of zeroes, and except for the
trivial zeros at integer negative values, all the rest seem to lie on
the line real(s) = 1/2. In other words, the real part of the zero is
1/2, but the imaginary part varies. The smallest such root has
imaginary part about 14 (and -14, since it's symmetric about the real
axis).

All known zeros (which includes tens of thousands of them) are on the
line real(s) = 1/2, but nobody knows for sure if they all are.
Riemann's hypothesis is that all are on that line.

If it's true, we will know a lot more about the distribution of prime
numbers, among other things. Much progress has been made recently,
using, surprisingly, results from the field called "random matrices."

The Riemann hypothesis is usually covered in your second or third year
as a mathematics graduate student, so if the above doesn't make sense,
perhaps this is why.

- Doctor Tom, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Imaginary/Complex Numbers

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