Riemann Hypothesis

**Whoever** · January 26th 2009, 03:26 AM

Hi folks.

I'm trying to understand the principles behind the Riemann Hypothesis better. Is there someone here that can help? I've read what I can on this but can't find an answer to some very basic questions.

My first question would be, what set of numbers is fed into the Zeta function to produce R's landscape in the complex plane? Or is even this a daft question?

Whoever

**mr fantastic** · January 26th 2009, 03:36 AM

Originally Posted by Whoever

Hi folks.

I'm trying to understand the principles behind the Riemann Hypothesis better. Is there someone here that can help? I've read what I can on this but can't find an answer to some very basic questions.

My first question would be, what set of numbers is fed into the Zeta function to produce R's landscape in the complex plane? Or is even this a daft question?

Whoever

Get this book: Amazon.com: Riemann's Zeta Function: Harold M. Edwards: Books

**Rincewind** · January 27th 2009, 05:23 AM

Originally Posted by mr fantastic

Get this book: Amazon.com: Riemann's Zeta Function: Harold M. Edwards: Books

Edwards is a good book but he uses an unusual version of the factorial function $\Pi(z)$ rather than the usual gamma function. Another decent book is Ivic's The Riemann Zeta-Function Theory and Applications.

In answer to the original post (after a fashion). To investigate the numerical landscape of the RH you need to find the roots of $\zeta(z)=0$ in the critical strip $0 < Re(z) < 1$ . The hypothesis states all the roots lie on the line $Re(z) = 1/2$ , and if you find just one root off the line (but in the critical strip) then you have disproved the hypothesis.

Proving it is a little trickier and left as an exercise for the reader.

**Whoever** · January 27th 2009, 10:09 AM

Thanks for the replies, but I'm afraid this is getting over my head already. Please talk to me as if I were a simpleton.

Let me try to be more clear about what I'm trying to understand. By the way, I've read Derbyshire, Du Sautoy and many articles. But they don't give me what I want. They start where I want to end.

I'm happy to treat the Zeta function as a black box.The first thing I need to understand is, what sequence of numbers are fed into this box to make Riemann's landscape come out the other end? Or does even this question betray a misunderstanding? I've asked it a number of times and never get an answer.

**Rincewind** · January 28th 2009, 02:45 AM

Originally Posted by Whoever

Thanks for the replies, but I'm afraid this is getting over my head already. Please talk to me as if I were a simpleton.

Ok. To try the simpleton suggestion $\zeta(z)$ is a function of a complex variable. So let's call $z = s +it$ where $s$ and $t$ are both real and $i$ is the imaginary unit, i.e. $\sqrt{-1}$ .

Also generally the value returned by zeta will also be complex. And we are interested in the zeros. That is the real and imaginary component both zero.

Now Riemann's hypothesis is that for $0 < s < 1$ , then all the roots occur for $s = 1/2$ . That is there are no roots elsewhere in that range.

So the numbers you would need to "feed to the black box" are complex numbers $z = s + it$ with $0 < s < 1$ and what you are looking for are returned value that are completely zero (real and imaginary components both zero). You can find infinitely many for the line $z = 1/2 + it$ , but no one has found one for $s \ne 1/2$ .

I hope this is at the right level. If not, I suspect you need to find out about complex numbers, functions and functional analysis. This isn't really a problem in number theory, despite the connection to the prime distribution.

**Whoever** · January 28th 2009, 05:33 AM

Originally Posted by Rincewind

Ok. To try the simpleton suggestion $\zeta(z)$ is a function of a complex variable. So let's call $z = s +it$ where $s$ and $t$ are both real and $i$ is the imaginary unit, i.e. $\sqrt{-1}$ .

Ok. I have the general idea, even if not an understanding.

Also generally the value returned by zeta will also be complex. And we are interested in the zeros. That is the real and imaginary component both zero.

Thanks, I learnt something from that. But what are s and t? Suppose I wanted to create R's landscape. What value for s & t would I start with and how are these two values related? How do I construct a sequence of complex numbers to feed into the box?

Now Riemann's hypothesis is that for $0 < s < 1$ , then all the roots occur for $s = 1/2$ . That is there are no roots elsewhere in that range.

Nope. I can't follow the notation here.

So the numbers you would need to "feed to the black box" are complex numbers $z = s + it$ with $0 < s < 1$ and what you are looking for are returned value that are completely zero (real and imaginary components both zero).

Ok, I think I get that. But which complex numbers are used? Any sequence of them?

You can find infinitely many for the line $z = 1/2 + it$ , but no one has found one for $s \ne 1/2$ .

Ok. So where s = 1/2 we find that the Zeta function outputs nontrivial zeroes only on the critical line, or that is the conjecture. Yes? No? But where s = 1/2 how do we assign values to i and t?

I hope this is at the right level. If not, I suspect you need to find out about complex numbers, functions and functional analysis.

I fear you may be right about this. But I'm still unsure why it should be so. If I have a function z = 1/2 + x/2, we can run x through the integers and plot a graph. Is this not all we are doing to create R's landscape?

Perhaps I could change the question a bit. If we could prove that all the nontrivial zeroes fall on the critical line, what would this tell us about the primes?

Many thanks for your replies. I realise this is probably a tedious discussion for you but I'm finding it helpful. I'm not actually a simpleton by the way, and have constructed a heuristic proof of the twin primes conjecture and also a spreadsheet which will calculate the prime sequence up to the limit of my version of Excel. All with no understanding of proper mathematics. So I find it intensely frustrating that I cannot grasp how R's landscape is created. I know I have some misconceptions about it but can't pin down what I'm misconceiving. Perhaps it's a lost cause, but I'd like to persevere for a while longer of that's ok with you.

Cheers
Whoever

**Rincewind** · January 29th 2009, 04:06 AM

Originally Posted by Whoever

Thanks, I learnt something from that. But what are s and t? Suppose I wanted to create R's landscape. What value for s & t would I start with and how are these two values related? How do I construct a sequence of complex numbers to feed into the box?

Ok the thing is to think of it as a function of two variables: s and t. Think of the formula for the volume of a cylinder $V = \pi r^2 h$ you can vary r or vary h and both independently effect the volume. It is the same with the zeta function. You can vary s or t independently.

Originally Posted by Whoever

Nope. I can't follow the notation here.

So the critical strip is where s is positive but less than 1. t can be anything.

Originally Posted by Whoever

Ok, I think I get that. But which complex numbers are used? Any sequence of them?

Basically we are interested in all the numbers where s is between 0 and 1 and t can be anything. There is no single sequence of numbers from which to build up the numerical landscape.

If you look here Riemann Zeta Function -- from Wolfram MathWorld you will see some surface plots of the zeta function.

Originally Posted by Whoever

Ok. So where s = 1/2 we find that the Zeta function outputs nontrivial zeroes only on the critical line, or that is the conjecture. Yes? No? But where s = 1/2 how do we assign values to i and t?

yep the conjecture is that for 0 < s < 1 then all the zeros, s=1/2. The trivial zeros are all outside the critical strip.

Originally Posted by Whoever

I fear you may be right about this. But I'm still unsure why it should be so. If I have a function z = 1/2 + x/2, we can run x through the integers and plot a graph. Is this not all we are doing to create R's landscape?

Perhaps I could change the question a bit. If we could prove that all the nontrivial zeroes fall on the critical line, what would this tell us about the primes?

There are many results which have been investigated and proven as true provided that RH holds. However the RH remains to be proved or disproved. One of the results is to give a better upper bound on the sum integral difference of the prime number theorem.

Originally Posted by Whoever

Many thanks for your replies. I realise this is probably a tedious discussion for you but I'm finding it helpful. I'm not actually a simpleton by the way, and have constructed a heuristic proof of the twin primes conjecture and also a spreadsheet which will calculate the prime sequence up to the limit of my version of Excel. All with no understanding of proper mathematics. So I find it intensely frustrating that I cannot grasp how R's landscape is created. I know I have some misconceptions about it but can't pin down what I'm misconceiving. Perhaps it's a lost cause, but I'd like to persevere for a while longer of that's ok with you.

As I said above. This of zert as a function of 2 variables which also returns two variables and maybe that will help with how you approach the problem. Cheers.

**Whoever** · January 29th 2009, 04:56 AM

Originally Posted by Rincewind

Basically we are interested in all the numbers where s is between 0 and 1 and t can be anything. There is no single sequence of numbers from which to build up the numerical landscape.

Aha.

If you look here Riemann Zeta Function -- from Wolfram MathWorld you will see some surface plots of the zeta function.

Thanks. Can't get the page to load at the moment but will try later.

yep the conjecture is that for 0 < s < 1 then all the zeros, s=1/2. The trivial zeros are all outside the critical strip.

Ok, got that.

There are many results which have been investigated and proven as true provided that RH holds. However the RH remains to be proved or disproved. One of the results is to give a better upper bound on the sum integral difference of the prime number theorem.

Ok. So RH bears on the distibution of primes. Yet I can see no connection between the numbers we input into the Z function and the primes. Where does this connection come in?

Thanks again.
Whoever

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