# Thread: Is the Riemann Hypothesis hard to understand?

1. ## Is the Riemann Hypothesis hard to understand?

I read a book about the Riemann Hypothesis, more of a book telling the story, but it explained in elementary form some of the math involved. But atleast at the very minimum, what areas of mathematics do you need to understand before you can truly understand what Riemann ment in his famous hypothesis? I mean as it was when he first postulated it, discluding all the mathematical subjects neccesary to understand the research done on the hypothesis in the modern time.

3. Riemann Hypothesis is really hard to understand , even for undergraduates .

You may say what so difficult ? The real part of all the zeros of zeta function in $\displaystyle (0,1)$ is $\displaystyle \frac{1}{2}$ , which is what mathematicians need to prove , but do you really understand the actual meaning behind this ? Why is that important to prove the zeros all lying on the critical line ? What can it be applied in various aspects ? It is hard to understand ...

4. Originally Posted by simplependulum
Riemann Hypothesis is really hard to understand , even for undergraduates .

You may say what so difficult ? The real part of all the zeros of zeta function in $\displaystyle (0,1)$ is $\displaystyle \frac{1}{2}$ , which is what mathematicians need to prove , but do you really understand the actual meaning behind this ? Why is that important to prove the zeros all lying on the critical line ? What can it be applied in various aspects ? It is hard to understand ...
I'm not disputing the fact that its a difficult subject. Its most certainly a complex concept (I couldn't resist the pun). I was more so curious as to the subjects neccesary to access an understanding of the hypothesis. Complex analysis, real analysis, number theory, what else? Pretty much the majority of advanced mathematics?

5. Originally Posted by mfetch22
I'm not disputing the fact that its a difficult subject. Its most certainly a complex concept (I couldn't resist the pun). I was more so curious as to the subjects neccesary to access an understanding of the hypothesis. Complex analysis, real analysis, number theory, what else? Pretty much the majority of advanced mathematics?
Those are really the only branches you'll need to understand the conjecture.

6. There is a neat interpretation on pg 268 in H.M. Edwards book titled "Riemann's Zeta Function".

It's called Denjoy's interpretation and goes like this:

Consider $\displaystyle M(x) = \sum_{1}^x u(n)$

It is known that the Riemann Hypothesis is equivalent to the hypothesis that $\displaystyle M(x) = O(x^{0.5 + \epsilon})$ for any $\displaystyle 0 < \epsilon$

Due to the fact that $\displaystyle u(n)$ behaves very unpredictably, the above hypothesis lends itself to the probabilistic interpretation that M(x) is just the number of heads minus the number of tails in a sequence of x flips of a fair headed coin.

Under this interpretation, it easily follows from the Central Limit Theorem that $\displaystyle M(x) = O(x^{0.5 + \epsilon})$.

While the above is obviously not a proof, its interesting in that it suggests the Riemann hypothesis follows more from some kind of unpredictability of $\displaystyle u(n)$ rather than some predictable property of it. It also begs the question of just how "unorderly" does a function need to behave in order to apply the Central Limit Theorem to it.

7. Originally Posted by simplependulum
Riemann Hypothesis is really hard to understand , even for undergraduates .
That is a mildly humorous sentence, because being an undergraduate is no qualification whatsoever. I would say that,

`(The) Riemann Hypothesis is really hard to understand , even for postgraduates'

would be a better line...

8. I recommend the book "Prime Obsession" by John Derbyshire.

It's the most accessible and enjoyable Mathematics book I've read. The way it's set out is that every even numbered chapter gives the math side of the story, and the odd chapters go in to the history.

You will need a decent grasp of complex numbers and analysis, but Derbyshire really does make it a book for everyone.

Give it a whirl people!