Interactive Online Tutoring Services

It seems to me there may be an easier way to express the zeta function: Z(s)=1/1^s+1/2^s+1/3^s…..using the ideas of concept sharing as it applies to a new geometry.

Concept Sharing and a new view of the Riemann hypothesis

Abstract: Here I introduce concept sharing. In uncovering extended space, I develop new ways of understanding the Riemann hypothesis.

Overlapping shadows:

An interesting notion can come about from something observed in nature, the overlapping of shadows. Consider a cup of tea placed on a table and lit with two lights from above, one from the left and another from the right, as illustrated below:

Three areas of shadows are formed. The one in the center is the overlapping of two shadows. 

The concept sharing of a number:

Numbers are exact concepts. In the above case, we can think of them as the number of shadows at the center. They have exact boundaries and some way of showing we have two there or three there, ect.

Then borrowing from the notion of overlapping shadows we should be able to hide numbers together and they would be “two hidden as one” as well. (concept sharing) if the mathematical objects represented by the numbers had the same boundaries, like the shadows at the center. 

Other than the further darkness of the overlapping shadows, we cannot see or imagine that there are two separate shadows there. Similarly with two numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.

Then to this end let us create another number dimension, a dimension of number of numbers. Let the usual case be that the number of numbers is only 1. But now let us expand into the next dimension and allow the number of numbers to be 2.

So for example with the number 1, let us take away the original number 1 (since we have another underlying dimension, we can do this) and replace it with two new numbers 1’(1) and1’(2). These are together like the two shadows but do not form one number.

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects hidden as one.

The objects are somehow different from each other. We give the two hidden objects two new numbers 1’(1) and 1’(2). 

In the case of mathematical objects there is no external way of telling how many objects there are, previously it was assumed it was only one. We can state how many we wish at the onset thus fixing a certain mathematical system. Then we need the concept sharing of a number to indicate how many objects we wish to be there

A new plane:

Points are also exact concepts. In the Euclidean plane they are places, with the notion of no extent, in the plane. We should be able to place two together using two new numbers 0’(1) and 0’(2) identifying that we have two points. (0 is indicating an origin)

An object of no extent placed together with another object of no extent, would still have no extent- but there could be two objects here, under another mathematical system. 

The two points 0’(1) and 0’(2) can be different by first uncovering a new place dimension, a place of places. This must already exist because there must be some way to have two points exist together and still be two points.

In a similar way as we uncovered the new number dimension (the number of numbers) we can uncover the new place dimension. 

Take the original point out (we can do this since we have a new underlying dimension of place, a place of places) and replace it with the two new points. This can be done for the whole plane of points.

That is, there is nothing special about the origin, so each point of the usual plane can be removed and we can replace it with a “sharing” of two points. So that we have a whole plane of doubled points co-existing with a plane of places of places. 

One of the new points can be fixed, while the other one is capable of “shifting” away in this new dimension of place. In this way these two can be different. Then all of the sharings in the new plane can become new origins-one point being fixed while the other point is capable of shifting away.

The Riemann Hypothesis:

One may imagine a type of gird with the first square being 1, the next being 1/2^2 the next being 1/3^2… if we use s=2 as an example. See pictures in the note below. The higher numbers of s can be seen by increasing the dimension. Yet there is always a plane possible with any dimension.

Since with concept sharing geometry there comes a place of places, in which places can vary, we may vary the distance as we choose to always make the zeta function defined. The zeta function can be continued into the extended geometry. Then there is no longer a need for analytic continuation. I can always make the grid into a 1 by 1.

So we can create a grid specific to the Zeta function defined in placement space.

Then we have that there are two types of number involved. A real part and an imaginary part.

I think this can be seen more primitively as a numbers which lead to a square with a positive area and numbers which lead to a square with negative area ie. the negative distance is i. These can be sharing space and then later moved to form the usual complex plane.

We can concept share two different numbers in the following way: (1(1)*(i(i) where * is a concept sharing of a concept sharing= ((1(i))((1(i)). But 1 and i have to be different. Before i is i, it has to be different from 1. Then let this be how the square comes about. We have to expand the zero-dimensionality of the concept sharing. The counterparts in higher dimensions can be j^3=-1 in three space, k^4=-1 in four space, etc. That is, the length can change depending on the dimension. This can be accomplished by a splitting of the line in to halves, thirds, fourths, etc. Since 1/2 of zero extent is still zero extent. Similar to twice zero extent being still zero extent.

Then it comes about that there is a line formed in the real part with a definite center which is clearly 1/2 of the whole, as I can’t go beyond 1/2 given the zeta function definition and the imaginary part can be any imaginary number that we choose. We create the complex plane by separating the imaginary part from the real part. This shows the solutions of the extended Zeta function to have a real part of 1/2 and a variable imaginary part.