## Archivo para la etiqueta ‘Riemann zeta function’

By means of interferences between prime functions this post shows how an asymmetry between complex conjugates non-trivial zeros inside of the critical strip appears in the Riemann Zeta Function when the prime harmonic functions have a different phase, which could challenge the Riemann Hypothesis while clarifying the relation between prime numbers and the Riemann non-trivial zeros.

(I’m going to try to summarize here the previous post to especially bold the case where an asymmetry between the complex conjugates non-trivial zeros of the Riemann Zeta function takes place, which could represent a counter example of the Riemann Hypothesis).

The Riemann Hypothesis states that all non-trivial zeros of the Riemann Z function have a real part one half.

Let’s verify it by firstly creating separate functions for every prime number, and then overlapping them on the function of the prime 3 to see if they create interferences on a cycle of the function 3.

To build such functions I drew two columns for every prime, a column for odd numbers and a column for even numbers. The cycle of each function is given by adding 2 times the value of the prime number to that prime itself, for example for the case of 3, we add 3+3+3 getting a cycle whose length goes from 3 to 9.

Then, let’s trace a dotted line from the valley of each function different of 3 to project them on the function 3. When those projected lines overlap the valley of the cycles of the function 3 no interference will happen. But when thy overlap an amplitude of the function 3 an interference is going to take place on the affected – number of the function 3.

The first cycle of each function different than 3 won’t interfere, and without computing such first cycle, no interference will take place every 3 cycles, because their valleys will overlap, those cycles will be divisible by 3.

Let’s see as an example the non-prime number 15 on function 3. The valley of the first cycle of function 5 overlaps the valley of the second cycle of function 3. That -15 point of function 3 will be a trivial zero. It’s a zero in the sense of the Riemann Z function because we reduce the value of the even +10 to ceros from function 5 to function 3, being located its ceros at the point where -15 is located.

Then, let’s check what happens with -25 on function 3. We see that the valley of the second cycle of function 5 interferes with the amplitude of the fourth cycle of the function 3. In that case, we have reduced to zeros the even +20 from function five to function 3, reaching that zero value at the valley where the point -25 of function 3 is placed. That point will be a non-trivial zero.

In the Riemann Zeta function trivial zeros are those that are located outside of the critical strip, while non-trivial zeros are those that are placed inside of the critical strip.

What is the critical strip in the above diagram? I have divided a continued strip into separate pieces, so we are considering a piece of the strip for each cycle of the function 3. The zeros that placed inside of a cycle of the function three, for example the zeros placed at -23 or -25, will be non-trivial zeros of the Zeta function, while the zeros that are outside of a cycle of the function 3 (acting as limits of the cycle’s length, for example the zeros placed on -15 or -21) will be trivial zeros.

As all the trivial zeros are going to be placed at the lowest point of the valley of function 3, we know they all will be related to non-prime numbers, as their emplacement will be always divisible by 3.

We do not know yet if the – odd numbers placed inside of the critical strip (inside of each cycle of function 3) are prime or non-prime numbers.

Let’s examine now if a non-trivial zero inside of the critical strip has or non has an imaginary part in the complex plane:

Here we take a section of functions 3 and 5, and we reduce to zeros the + 30 from function 5 to the point -35 of function 3. Function 3 and 5 are harmonic or orthogonal. The lowest point of the valley of the cycle of function 5 is going to be a non-trivial zero because it’s placed inside of piece of the critical strip that exists between the limits of -35 and -39.

We see that the non-trivial zero coincident with -35 has a 1/2 real part and an 1/2 imaginary part existing at the point of the intersection between the positive trajectory of the function 5 going from -35 towards +40, and the negative trajectory of the +36 of function 3 when going towards -39.

The odd number -37, that is not affected by the interference, will be prime because there won’t be a non-trivial zero placed in that position, the non-trivial zero is projected on the imaginary axis related to the real -35.

But what will happen when inside of the critical strip there are two consecutive non-trivial zeros?

Both non-trivial zeros should have a 1/2 real part and an imaginary 1/2 part, as they have. And it seems they should be complex conjugate, having a mirror symmetry between them. But in the above case we see the mirror symmetry is broken when it comes to one of the imaginary parts.

The imaginary axis of the 1/2 imaginary part related to the 1/2 real part of 187 would be located in some non integer number point between 183 and 185.

If the twin non-trivial zeros must be complex conjugate to accomplish the Riemann hypothesis statement, then this would be a counter-example of the RH.

(On the above picture, another observation would be that, interestingly, the distance between 185 and i? seems to be coincident with the distance between 183 and 185, 185 and 187, or 187 and 189 on the real axis).

Below I attached another case of this asymmetry caused by the displacement of the imaginary axis, The diagram represents the non-prime number 221 and the prime 223 inside of the critical strip whose limits are determined by 219 and 225. The valley of prime function 13 interferes with the amplitud of prime function 3 causing a non-trivial zero at the point -221.

Here, the imaginary axis related to the real 1/2 non-trivial zero of 221, is not placed on 223 but on a non integer number between 223 and 225.

It seems the asymmetry of the imaginary axis appears because the amplitudes of the cycle of the function 13 (at points 210 and 234) are not convergent with the amplitude of a cycle of the function 3, that is to say, because they have a different phase.

And what does to be a different phase mean when it comes to even and odd numbers? That is something I still have to think about, i guess it depends on the way those numbers are composed and the symmetry they represent, and on the starting point of their functions.

But another way of thinking about the real and imaginary axis would be to consider that the straight line we can trace from -35 to +36 is the real axis, and the perpendicular line that divides in two equal parts the segment between 35 and 36 is the imaginary axis. In that way it could be said the real part of the non-trivial zero is 1/2.

This another interpretation seems to fit better with the Riemann Zeta Function, because it seems Riemann was not naming his trivial or non trivial zeros because of the interference they cause to determine the prime quality of a number (he did not know what was really the relation between his Zeta function and the primes distribution); it seems he was calling non-trivial to the zeros that are in that critical line – which is a line in the complex or imaginary plane – having 1/2 part in the real axis when two harmonic functions converge. That would correspond in these diagrams with the imaginary point where the intersection of the functions 3 and 5 (or another interfering prime function different than 3), when a function different than 3 has its valley on (which for us is the non trivial zero actually) an odd number where the function 3 has not a valley.

In this sense, I guess that for Riemann the trivial zeros would correspond in these diagrams to the points of intersection between functions 3 and 5 (in this case) that are not in the critical line but closed to an even or odd number.

I have put below more clearly, I hope:

On the above picture, we can see how there is a mirror symmetry actually and the non trivial zero that is placed at the number -25 does not have its imaginary extension at the next point of intersection of the functions 3 and 5 (point that I think would be for Riemann a trivial zero) but before, at an intersecting point which is just on the critical line.

But for me the actual non trivial zeros, an that is the part that connects the Riemann Zeta function with the primes distribution is the zero point placed on an odd negative number that is the valley (coming from the amplitude of an even positive number) of a function different than 3 when the function 3 does not have a trivial zero point, a valley then.

But it seems that when the length of the prime function that converges with the function 3 is very large, the convergent point is going to be outside of the critical line (considered it in the way represented on the above picture), and that the position of the imaginary point that represents the converging point is going to be closer and closer to the real odd axis as the length of the further prime functions become larger and larger. Would it imply that is going to be a point (a point at the infinite?) where the imaginary plane is going to be just on the real odd axis? And what would be the spatial difference between the trivial and non-trivial zeros then?

Anyway, I think the most important thing is not to confirm or refute the Riemann hypothesis, but mainly to understand the meaning of the Zeta function, because our mathematicians develop algebraic tools making calculations which meaning is mainly unknown for themselves. Mathematicians currently do not know what they even need to confirm or refute the Riemann hypothesis because no one understands what’s the Riemann zeta function actually about. What’s the Riemann hypothesis looks like in the real world? Some advances were made years ago when a relation between atomic spectral lines and the prime numbers distribution were found out.

Another way of representing this approach would be these:

The real part of the imaginary non-trivial zero point (where the intersection between functions takes place) that is relevant for us is the Real odd number 35. When getting a non-trivial zero in the complex plane, what is necessary is to determine, in the real axis of the odd numbers, what is the real odd part of that imaginary zero.

On the other hand, as Riemann himself was aware, it’s known that the Riemann hypothesis suggests a deep link between the Riemann Zeta function and the prime numbers distribution that not clearly understood so far.

I think this unorthodox approach that uses prime functions interferences would let visualize what’s actually happening with the Zeta function – that is currently being used in a blind way as an abstract calculation tool – clarifying its link to prime distributions.

In this sense, When there’s only a non-trivial zero in the critical strip piece we are interested in, the affected odd number will be non-prime, and the another position will be occupied by a prime number; when there are two non-trivial zeros inside of the critical strip piece, its two position will be occupied by two consecutive or twin non-prime numbers; and when there are no non-trivial zeros inside of a piece of the critical strip that will mean that both odd consecutive numbers placed there will be prime twins.