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The Riemann Hypothesis - one of the outstanding unsolved problems in math

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Re: The Riemann Hypothesis - one of the outstanding unsolved problems in math

#11

Post by hsmyers »

freyrs3 wrote: post #6
btilly wrote: post #5

I can't ever read a claim that the Riemann Hypothesis is unsolved without putting a small asterisk beside that claim. See http://www.lrb.co.uk/v26/n14/karl-sabbagh/the-strange-case-o... for why.

The proof has been checked and it contained errors. See: http://mathoverflow.net/questions/38049/what-exactly-has-lou...

So did the work of Andrew Wiles--- and instead of slamming the door on the effort support, suggestions and help came from all around until the effort was successful. Yes Andrew was probably a much more well-liked player, but that should have no bearing on scientific inquiry. The subject mathematician has a track record and is not considered (even by his obvious enemies) a crank. Makes you wonder what else is being suppressed...

Re: The Riemann Hypothesis - one of the outstanding unsolved problems in math

#14

Post by btilly »

btilly wrote: post #7
gulbrandr wrote: post #2

"We would like to see the prime numbers distributed as regularly as possible." Why? I don't understand why mathematicians seek regularity in prime numbers so much.

In this case it is statistical regularity. :-) The primes act very much like they were a random set of numbers with n having probability 1/log(n) of being prime. (Mathematicians always mean "natural log" when they say log.) Except, of course, that this is obviously false. Because, for example, there is only one even prime. It is more accurate to say that the primes act very much like they were {2} plus a random set o…

When I woke up and thought about this problem again, I realized that I didn't explain one detail. Why should (pi(n) - li(n))/sqrt(li(n)) follow a standard normal distribution?

Here is why. pi(n) is the sum of a bunch of many independent random variables, each of which looks like a flip of a biased coin. If you pick the right version of the central limit theorem (see http://en.wikipedia.org/wiki/Lyapunov_condition for the right version), it is clearly going to be approximately normal.

Its average will be the sum of the averages, which is 1/log(2) + 1/(log(3) + ... + 1/log(n) which is going to be li(n) plus O(1). If n is large the li(n) term dominates.

Its variance will be the sum of the variances of the individual random variables. The variance of a coin with probability x of turning up 1 is x(1-x). Therefore the sum of the variances is (1/log(2) - (1/log(2)2) + (1/log(3) - (1/log(3)2) + ... + (1/log(n) - (1/log(n)2) which is (modulo a O(1) error) the integral from 2 to n of 1/log(x) - (1/log(x))2 which is li(x) - O(n/log(n)2), which is dominated by the li(x) term.

Therefore the average and variance of pi(n) are approximately li(x). So (pi(n) - li(n))/sqrt(li(n)) should have approximately a standard normal distribution.

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