Riemann Hypothesis

[boooeee]

I just finished reading "Prime Obsession" by John Derbyshire about the Riemann Hypothesis. I thought that this was a very well written book that did a good job of describing the history of the Hypothesis and the failed attempts and near misses at proofs.

I also thought that the book did a very good job of getting into the nitty-gritty details of the mathematics underlying the hypothesis.

That being said, there seems to be a huge, glaring omission in the book that is driving me crazy. After the author goes to great pains to demonstrate mathematically the link between the Riemann Zeta function and the distribution of prime numbers (no small feat for a book meant for the general public), he seems to forgotten that his book was about the Riemann Hypothesis, which is:

All non-trivial zeroes of the Riemann Zeta function have real part equal to one half.

So, I am hoping somebody out there in the forum can answer the following questions for me:

- Why did Riemann think that all the non-trivial zeroes had real part one half?

- What are the mathematical consequences of the Riemann Hypothesis being true? In other words, what is so significant about the zeroes having real part one half (and not one third, one fourth, etc.)? How would our knowledge of prime numbers change if some zeroes did not have real part one half?

 

[Tez]

Riemann computed the first fiew non-trivial zeros, and found tem to have real part 1/2. I suspect what he did then was examine the consequences of all zeros having real part one half, and found that the prime number theorem (which was not a proven theorem IIRC when he was alive, but the conjecture of it was known) can be made much tighter (i.e. one can bound the error) if the conjecture of all zeros having real part 1/2 was true.

I've no idea what the consequences would be of having zeros off the line x=1/2. However it would presumably imply drastic things for the distributions of primes. For example - the prime number theorem, one of the strongest theorems on prime number sitribution, follows from the fact that you can prove no zeros lie on the line x=1. Presumably proving that no zeros lie on other values of x would provide strong bounds on the distribution of prime numbers...

 

[aerocontrols]

People interested in the book may also be interesting in hearing Derbyshire sing the song that can be found in the appendix of the book.

Or not.

I thought it was funny, though.