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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?


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/   
    
Associated Topics:
High School Imaginary/Complex Numbers

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