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1.
Without question, THE most important unsolved problem in all of mathematics. The hypothesis, put forth by Bernhard Riemann in 1859, states that ALL non-trivial zeros of the Zeta function have a real part of one-half. Another way to put this is that the distribution of numbers with an odd number of prime factors vs. those with and even number of prime factors is roughly 50/50. Anyone who can prove the Riemann Hypothesis without the aid of a computer will win a $1,000,000 prize. A great deal of mathematical theorems rest on the assumption that this hypothesis is true. If anyone can find a non-trivial zero of Zeta with a real part not equal to 1/2 will destroy a lot of mathematical work, and become very famous.
The Zeta function = 1 + 1/2^S + 1/3^S + 1/4^S .... to infinity. The Riemann Hypothesis states that Zeta = 0 only when the real part of S = 1/2 (non-trivial zeros on the critical line). Has eluded a proof for almost 150 years.
by Ignatz April 28, 2008