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

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

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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?

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

Nope. I can't follow the notation here.

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So the numbers you would need to "feed to the black box" are complex numbers $\displaystyle z = s + it$ with $\displaystyle 0 < s < 1$ 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?

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

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?

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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