(to the tune of "Sweet Betsy from Pike")
Where Are The Zeros of zeta of s?
G.F.B. Riemann has made a good guess:
"They're all on the critical line", stated he,
"And their density is one over two pi log T."
This statement of Riemann's has been like a trigger,
And many good men, with vim and with vigor,
Have attempted to find, with mathematical rigor,
What happens to zeta as mod t gets bigger.
The efforts of Landau and Bohr and Cramer,
Hardy and Littlewood and Titchmarsh are there.
In spite of their effort and skill and finesse,
In locating the zeros there's been no success.
In 1914 G.H. Hardy did find,
An infinite number that lie on the line.
His theorem, however, won't rule out the case,
That there might be a zero at some other place.
Let P be the function of pi minus Li;
The order of P is not known for x high.
If square root of x times log x we could show,
Then Riemann's conjecture would surely be so.
Related to this is another enigma,
Concerning the Lindelof function mu sigma,
Which measures the growth in the critical strip;
On the number of zeros it gives us a grip.
But nobody knows how this function behaves.
Convexity tells us it can have no waves.
Lindelof said that the shape of its graph
Is constant when sigma is more than one-half.
Oh, where are the zeros of zeta of s?
We must know exactly. It won't do to guess.
In order to strengthen the prime number theorem,
The integral's contour must never go near 'em.
Tom M. Apostle
I did my final year project on trying to explain the Riemann Hypothesis - came across some really nice maths...
I think any proposed proofs are given a 'scrutiny' time of 2 years before it'd be declared solved, and the prize awarded.
It was my interest in number theory that led me to the problem, and I've always referred to it as a number theory problem because I hated complex analysis
There's a huge number of number theory 'theorems' that begin with the assumption that Riemann's Hypothesis is true - so if it isn't, that's quite a number of conjectures that will have to be thrown out or adjusted...
Ramanjuan worked on the Hypothesis with Hardy, but the only really remarkable thing in relation to it was the amount of stuff he REdiscovered, that had already been known in the west for some time.
Hardy and Ramanjuan made inroads into Goldbach's weaker conjecture form assuming that the Riemann Hypothesis was true, although I never really got to look into how that worked.
And interstingly if you prove the Riemann Hypothesis is impossible to prove - that means that it must be true Because if it's not true - there's a zero off the line - which would be provable
It'll be interesting to see how this develops... Keep us updated!
Maybe if TPH has more news from his tutor he will post it here. We'll have to wait and see.
I read somewhere that if it is solved, then all cryptic codes like internet encryption could be easily decrypted, making online security(internet banking, online shopping, etc) a thing of the past. Is this really true?
(I am not sure about this because this area of math never interested me). There happen to be faster primarility tests which involve the Hypothesis. The proof behind these tests working all the time use the truth of Hypothesis. Which means if the Hypothesis is true these primality tests would get faster. But that will not give us a way to factorize large numbers. That is not what the problem is about.
It's said that Riemann's Hypothesis was a response to a paper by Gauss on the density of prime numbers and what it approached, Gauss' results became known as the Prime Number Theorem and Riemann's hypothesis related mainly to the error term and proving that it was in fact the best way to represent the error. He did some of the work through complex analysis.
There was an episode of NUMB3RS a season or two back where a brilliant mathematician(played by Neal Patrick Harris) was secrectly working on solving the RH. Some 'nare-do-wells' decided they would kidnap his daughter until he showed them how to use it to decrpyt and break into sensitive areas and steal money. Of course, in the end, he hadn't actually solved it and the bad guys were apprehended.