About interesting convergence of Riemann Zeta Function

[cryptist]

Hi, I was playing with Riemann zeta function on mathematica. I encountered with a quite interesting result. I iterated Riemann zeta function for zero. (e.g Zeta...[Zeta[Zeta[0]]]...] It converges into a specific number which is -0.295905. Also for any negative values of Zeta function, iteration results the same number.

I searched on internet, but I couldn't find any thing about it, besides one review text.

What is the meaning of that number? for s≤0, Zeta iteration converges to a number. Is that something important?

On the other hand, a lot of great mathematicians dealed with Riemann zeta function and I'm sure they have reached this result, and since nowhere mentions that, it may be too obvious or trivial. But I'm just curious about it. Can someone inform me?

 

[jedishrfu]

here's a Riemann visualization you could check out:

Did mathematica complain about the s<0?

 

[cryptist]

Actually, I mean Zeta iteration. Zeta of Zeta of Zeta of ... Zeta.

for s≤0, when I iterate Zeta, I am reaching same value.

 

[jedishrfu]

okay. are you trying to do some sort of fractal?

 

[cryptist]

No, I'm not trying to do anything. I saw an interesting thing when I played with Zeta function on mathematica. I stated everything related with my question, in my first post.

 

[jedishrfu]

fractal generation works like this: start with a point in the complex number plane and count the iterations over a given equation until it no longer converges(?):

http://en.wikipedia.org/wiki/Julia_set

there's one example fc(z) = z^2 + 0.279 shown midway through the article under quadratic polynomials

anyway that's all I can think of.

 

[lurflurf]

According to http://www.wolframalpha.com/
input
FindRoot[x - Zeta[x] == 0, {x, -0.305495, -0.287253}, WorkingPrecision -> 27]
output
x ~~ -0.295905005575213955647237831...

I do not know why that would be particularly interesting, more so than
Solve[x == Cos[x], x]
x~~0.73908513321516064165

As for negative values some representations of zeta may have trouble with negative values, but that is due to that representation, not the function.

For example a bad representation would be
$$\zeta (x)=\sum_{k=1}^\infty n^{-x}$$

 

[Vargo]

Just to clarify what lurflurf is saying (in case you didn't pick up on his point), the number you are getting is a "fixed point" of the zeta function. That is, it satisfies
Zeta(x) = x.

Oftentimes, if you iterate a function, you will approach a fixed point. The reason is the Banach fixed point theorem:

http://en.wikipedia.org/wiki/Banach_fixed-point_theorem

It is also called the contraction mapping theorem. I can't be sure whether this is the real reason because I don't know if the Zeta function really is a contraction mapping (on this particular subset of its domain), but perhaps someone else will clarify this.

 

[cryptist]

Just to clarify what lurflurf is saying (in case you didn't pick up on his point), the number you are getting is a "fixed point" of the zeta function. That is, it satisfies
Zeta(x) = x.

Oftentimes, if you iterate a function, you will approach a fixed point. The reason is the Banach fixed point theorem:

http://en.wikipedia.org/wiki/Banach_fixed-point_theorem

It is also called the contraction mapping theorem. I can't be sure whether this is the real reason because I don't know if the Zeta function really is a contraction mapping (on this particular subset of its domain), but perhaps someone else will clarify this.

Thank you Vargo for this enlightning explanation.

 

[RamaWolf]

x = -0.295905 is the first negative fixed point of the Riemann zeta, but it is not the only one: between x = (1.83 and 1.84) we find the only positive one and for example,
between x = (-24.0 and -24.01) or x = (-36.0 and -36.0000000001) we find others.

Proposition: We have fixed ponts of the Riemann zeta at x = - 12 * i - d
(for integral i > 1 and d a small (not constant) quantity)

But we have to remeber: The Riemann zeta is defined for complex values s and we are just talking about the case of Im(s) = 0

 

[Jeff Cook]

This fixed point, as with the others, seem to me to be mean values of the conditions imposed on them. By taking the zeta of the zeta of the zeta (or any other function) will eventually converge to some constant in most cases. However, in every occurence one opens up more possible solutions. Take the mean value of the cosine function for all natural numbers. Eventually as n maps to infinity, the average becomes zero. Problem is there are an infinite number of functions that converge absolutely...map to zero. In terms of the zeros of the zeta function, the roots, there are an infinite number of them and numerical evidence suggests that they are irrational numbers. Thus, by confining such numbers to a region where an infinite number of arguments converge on the same limit suggests the end result of amounting to less information in the end. What I would do is if I really felt there was some significance to a constant that I encountered ad hoc, I would attempt to express it in closed form, like phi = (Sqrt{5}+1)/2. If numerous serious attempts could not present such a solution, then I would probably make note and move on until the constant came up elsewhere. Important constants such as pi, e and the rest came from a careful study into the proceedure at hand, which is why the platonic solids and physical phenomena are useful in comparing mathematical solutions (especially in study of a Riemann Operator)...as we know then what the real problem at hand is. Good luck and keep calculating!