The Riemann Zeta FunctionDate: 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/ |
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