Ok, back to the der Veen and de Craats version of the Riemann functional equation.
z( -x ) = ((-2 * x!) / (2*pi)^(x+1)) * sin(pi*x/2) * z(x + 1)
What this states is that if you have the zeta(x) value for any complex x (x = a + bi) then you can get zeta(x + 1), and from there apply the functional equation to define zeta(-x).
This extends the zeta function to the left hand side of the complex plane (a < 0). Combining the functional equation with eta(x) and the original version of zeta(x) for x>1, we now have complete coverage of the complex plane everywhere except for the pole point x = a = 1.
One interesting feature here is that the mirror relationship between -z and z + 1 gives a center of rotation at x = 0.5.
c = B + (A – B)/2 = -z + (z + 1 – (-z))/2
c = -z + (2z + 1)/2
c = -z + 2z/2 + 1/2
c = 0.5, for any z.
I’ve mentioned before that the zeta(x) function converges to 1 for large x, meaning that there are no zeros for x>1.
And, the functional equation relates -z to z + 1, so again, there are no zeroes for x < 0, except for the trivial zeroes at x = -2, -4, -6, etc, because of the sin() component. (If there were any other zeroes to the left of x = 0, then there would have to be matching zeroes for x > 1, and there aren’t.)
That just leaves what is called the “critical strip,” for x = a + bi, 0 <= a < 1, which is covered by the eta(x) function.
Given the nature of the complex plane, if we know the value for x = a + bi, then we also have the values for a – bi, -a + bi and -a – bi. Which means that if zeta(a + bi) = 0, then zeta(a – bi) = zeta(-a + bi) = zeta(-a – bi) = 0.
That is, if there is a zero within the critical strip, it will have 3 more matching zeroes to go with it.
What Riemann did with his hypothesis was minimize the number of possible zeroes, by saying that all of the non-trivial zeroes lie on the critical line x = 0.5. Or, instead of having 4 zeroes for any given a + bi combination, there are only 2 (a + bi and a – bi).
So far, no one has been able to prove OR disprove this statement. Billions of zeroes have been found by using computers, and they’re all on the critical line. But, that’s not a proof. If even just one zero exists that is not on the critical line, then the entire hypothesis collapses, as do all the other theorems that are based on it.
Ok, so that’s the basic concept of the zeta function. We have the regular version,
z = 1 + 1/2^x + 1/3^x + 1/4^x…
For x > 1 across the complex plane.
We have the eta(x) function,
eta(x) = 1 – 1/2^x + 1/3^x – 1/4^x…
zeta(x) = eta(x) / (1 – 2^(1-x))
for 0 < x < 1 along the complex plane.
And Riemann’s functional equation above, for x < 0.
If we want to find the non-trivial zeroes, we need to use the eta(x) plus the conversion formula.
In order to follow the examples in the book, I’ve been playing around with Wolfram Alpha. It’s a pretty decent online math graphing tool, but there are some inconsistencies between the version online now and what der Veen and de Craats apparently used (dVdC say that the “|” character is used to calculate absolute values, i.e. “plot |x| for x = -10 to 10”. But that doesn’t work for me. I have to use “plot abs(x) x = -10 to 10). Most of the inconsistencies are minor, but it still takes time to figure them out.
The main thing is that because Wolfram Alpha already has the zeta function implemented, it’s very easy to see right away what it looks like near the origin. If we plot zeta(x) for (-30 -30i) to (30 + 30i), we get two graphs, one of the real component of the results and the other for the imaginary component.
Note that the results for both components get clipped at z = 1000. The important part of the complex plane is that strip between 0 and 1, where the results weave around z = 0, but that’s getting swamped out.
Plotting the absolute value of the above formula, we get the combined magnitudes of the real and imaginary components, but clipping is now occurring at 10^9. No zeroes visible from this altitude.
The next step is to concentrate only on the critical strip, and to try to include at least one of the zeroes. Remember, the zeroes themselves are on the critical line, x = 0.5.
And the absolute value of the plot as well. Note the pole at x = 1.
We can try plotting just the critical line, and to find the zeroes we need to see where both the real and imaginary components cross the x-axis at the same time.
Or, we can look at the absolute value of the graph, where the zeroes are the points where the line touches down on the axis. Regardless, the mirror nature of the complex plane is visible here. So, we only need to find the zeroes for y > 0 and we automatically get the matching zeroes for y < 0 at the same time.
This just leaves the graphs of the trivial zeroes.
As x goes more negative, the results get much bigger faster, swamping the y-axis crossings. So, I’m only showing the first few trivial zeroes, for x = -10 to 0.
While I’m at it, I might as well include a little of the gamma function. I’ll just use the part closer to the origin to show the best detail.
Next time: Tying the non-trivial zeroes to the Prime Counting Function