zetafunction-more than a billion-(the imaginary value of s)-very short

[AkiraBergman]

Hi Thomas, what is the number of vectors here?

[TRAJEKTORULM]

Hi Akira,

the imaginary value is about 1 001 000 698, should be 1 billion and something(in german: 1 Milliarde"). So, we can determine the vector in the middle of the last swung segment(a cornu-spiral, or clothoid spiral), more math is in an article of carl erickson about the zetafunction as exponential sum, i just learned to know some months ago. The angle between two consecutive numbers is determined by taking the difference of the according values of n and n+1.

[AkiraBergman]

Thanks for the reply, bur I was asking about the the number of iterations. To what maximum number do you iterate the sum? By the way, congratulations for the original work.

[TRAJEKTORULM]

one can only hardly say, the sum we get by iteration. But it is right, first we start at zero, add the first vectir (1/1^s), come to 1 on the x-axis, then add the second vector(1/2^s) and so on, the number of iterations, we do this is just the same, as the number of visible vectors. To make sure, to end at the center of the last "knot" or whirl, i took the value of iterations about Im(s)/pi, near 300 000 000 times. Reading the number i wonder, the computer made it.

[TRAJEKTORULM]

Geometrically, because the logarithmic function is the integral of the function 1/x(the hyperbel), we can say very accurately, that this difference is just about 1/n. This value 1/n we have to multiply with the imaginary value of s, in this case the 1billion and.. to get the angle between two consecutive vectors n and n+1. In the middle of the last segment, in the case of the zetafunction, the angle between the two vectors has to be 2pi, because then,

[TRAJEKTORULM]

has to be 2pi, because then, they show into the same direction, the resulting sum will go quite straight forward through the complex plane. By this, we can say, that for the vectors in the middle of the last segment we know: Im(s)*(1/n)=2pi. This gives us just n=Im(s) /2pi, which happens to be quite the number of zeroes to that value,

[TRAJEKTORULM]

... or how often you could roll the unit circle along the y-axis to get to the Imaginary value of s. In this case it will be 1001000698 / 6.28.. vectors, quite a lot. To determine the vectors to the center of the last segment, which will visit zero from time to time, the angle will be just pi, the vectors become opposite-directed, a new and last spiral will begin(for the center the number of vectors is about 1001000698/pi, (where is my calculator or at least a pencil?),

bets wishes,

thomas

[TRAJEKTORULM]

p.s.:please see also the book, if you want to get the mathematical aspects more exactly(but it is no easy thing to understand i had to accept and resign), of mendes france, ortega 24024 is talking about in his comment to the video zetafunction-the longplay

[TRAJEKTORULM]

the values of the length of the vectors and the angle between them is connected and related and this is the reason, why the zeroes shine up only for values of s with real value 0.5. The length of the vectors will be 1/sqrt(n) and the angle between them proportional to 1/n. I have tried to explain it more in detail in a little book, not so cheap, 19€, at book on demand. Sorry, it´s in german-If somebody would be interested, i could translate a short version in english.

So far-keep well

[TRAJEKTORULM]

a lot of now´s and but´s.

maybe it is easier to determine, where the zeroes of zeta(s) will be with this chain of vectors, which is without too much secrets. But the relation between the angle between the subsequent vectors is proportional to 1/2, 1/3 and so on because the angle between the vectors is the difference of the logarithm of the two vectors n-1 and n is gratefully ~1/n, and the length is 1/n^s.

[TRAJEKTORULM]

Now, it is always said: dding all these vectors, this chain will never come to one point, it always is carving through the complex plane, it"diverges". (Whereas the functional equation will give us a definite result). But- the chain diverges really, but in a very peculiar manner-the last segment ends up in a spiral, whose center marks the value, we get by the functional equation. The video shows the patterns of zeta(s) and its a bit more than just simply "diverging" and better visualized?Thomas

[TRAJEKTORULM]

continuing: 1/(2^s) for example means: we determine the value 2^s (with s a complex number, the result is a new complex number 2^s. This is a point in the complex plane, as well as the vector from zero to the point 2^s. Alas, its a vector. 1/(2^s) is the result of dividing 1 (as complex number 1+0*i, 1 on the x-axis) through this vector. Now, this result is also a vector. Now adding all these vectors, one after the other, we get an nearly equivalent form of the zetafunction

[TRAJEKTORULM]

usually the zetafunction(zeta(s)) is calculated using the functional equation, which is quite hard to understand and by this few(not me) do know, why the values of zeta(s) are as they are and where the zeroes (when the value of the function is zero). s is a complex number. This value determines the value of zeta(s). But-the zetafunction, examined by Euler, is originally determined as the value, we get, if we add 1/1+1/2+1/3... . And the complete zeta(s) is: 1/(1^s)+1/(2^s).. . see next comment