*** Riemann Hypothesis Still Unproven ***
~ Days after media reports emerged that an Indian mathematician had proven the Riemann Hypothesis, Clay Mathematics Institute, the grantor of the Millennium Prize, rejected the claim. The Reimann Hypothesis has remained unproven for over 160 years.
~ The Riemann Hypothesis (RH) has been the “holy grail of mathematics” since it was first conjectured in 1859. It has been called "the most famous unsolved problem in all of mathematics.”
~ To understand why RH matters, you must understand prime numbers. Recall your elementary school math teacher describing them as numbers that can only be divided by themselves and one, and that’s true, but that’s not all they are.
~ Any integer you can think of can be split into a unique set of prime factors.
~ To illustrate why this is an exciting question, consider how many primes are between zero and 10: four.
~ Now think how many there are between zero and 100: 25.
~ Between zero and 1,000, there are 168 prime numbers, and between zero and 10,000, there are 1,229.
~ So each time we increase the size of our interval by a factor of ten, the amount of it that is given over to prime numbers goes from 40 % to 25 %, to 16.8 %, to 12.29 %. In other words, primes are getting “rarer”.
~ By 1793, when he was 16 years old, Gauss had figured out the average % is approximately x/log(x).
~ Bernhard Riemann, a student of Gauss, found a pattern in the "frequency" of prime numbers. He found them to follow a closely followed pattern that could be explained with a function, which he called the Riemann zeta function.
ζ(s) = 1 + 1/2s + 1/3s + 1/4s + ...
~ RH that all attractive solutions of the equation ζ(s) = 0 lie on a specific vertical straight line.
~ That’s it! RH states that “The real part of every nontrivial zeros of the Riemann zeta function is 1/2”.
~ So much for the “average” behavior, but what about those “fluctuations” Gauss mentioned?
~ Well, those are related to the zeta function – and this is where Riemann comes in.
~ Bernhard Riemann, a student of Gauss, is not bad for a guy who did formal schooling until he was 14.
~ In his short but impressive life, he only wrote one paper on number theory, but it was a doozy.
~ In 1859, Riemann submitted a now-famous paper titled “On the number of primes less than a given magnitude.”
~ But this is not the end of the Riemann story.
~ In the late 18th century, the two legendary mathematicians Gauss and Legendre began, independently of one another, to study prime numbers.
~ But they had decided to approach the concept in a new way: they were looking at the "density" of the primes – the answer to the question “How many prime numbers are
expected in the range between 0 and n?”
~ As of late, no one has solved the Riemann Hypothesis in over 160 years, as well as the density aspect of the problem. See the post image.
--- B. Noted
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