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September 12th, 2011, 11:47 AM   #1
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Do you believe the Riemann hypothesis?

I'm fascinated, as many, by this conjecture that has been found to be true for the first ten trillion zeros that all lie on the critical line Re(z) = 1/2.

A disprove of the hypothesis, by proof or by finding one zero off the critical line, would mean prime numbers are more randomly distributed than we believe now.

According to Fields medalist Enrico Bombieri, "The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers".

Prime numbers are, like digits of ?, unpredictable and do not follow as it seems any rule - in the small scale. And in mathematics stating that is like saying that they follow all the rules. Gauss found the distribution in the large of prime numbers has an intimate relationship with the natural logarithm and subsequently e. The famous prime number theorem and its even better estimate, its integral Li(x).

Gauss' believed his prime number dice was fair. Statisticians regard a coin fair if after n tosses no side falls more than ?n of the time. That critical line Re(z) = 1/2? If it is true it becomes the exponent on the big O notation for prime distribution.
If the Riemann hypothesis is true, the nature's prime number dice is fair.

Apart from psychological reasons that a proof of the RH would bring great joy to the mathematician's community, it could with the advent of quantum computing destroy the safety of the current public-key cryptography and RSA. And that is because the infinity, as Hardy proved, of the zeros of Riemann's zeta function R(x) give the exact error correction for the best estimate for ?(?), the prime counting function.

So do you believe the Riemann hypothesis to be true? And if not, what is a good reason for it to not be true?
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September 12th, 2011, 02:07 PM   #2
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Re: Do you believe the Riemann hypothesis?

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Originally Posted by Eureka
So do you believe the Riemann hypothesis to be true?
Yes.
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