I have commented several times on the true significance of the zeta zeros.

Now once again from my own perspective, there are in fact two complementary sets of these zeros which I term Zeta 1 and Zeta 2. So Zeta 1 refer to the Riemann (non-trivial) zeros. The Zeta 2 by contrast refer to the various roots of unity (excluding 1, which is common to all roots.

Now once again the significance of these roots is that they enable seamless conversion as between the Type 1 and Type 2 aspects of the number system.

As we have seen when the Type 1 is associated with the quantitative (analytic) aspect of number behaviour, the Type 2 is then associated, in complementary fashion, with the qualitative (holistic) aspect.

So essentially the zeta zeros enable us to convert from Type 2 to Type 1 format, and equally from Type 1 to Type 2 format.

Without this facility we would have no reason to believe in the consistency of number operations from either the (recognised) quantitative or (unrecognised) qualitative perspectives.

However rather like the situation in physics, which conveniently breaks down into macro (relativistic) and micro (quantum) aspects, it is similar in the consideration of numbers.

So from one perspective, we can view each prime as composed of a unique group of natural number members (in ordinal terms).

From the other perspective, we can view the natural numbers as composed of unique groups of prime members (in cardinal terms) .

So from the first perspective we examine the micro nature of each prime (through its natural numbered ordinal members).

From the second perspective we view the macro nature of the natural number system (through its cardinal prime members).

Now from one perspective (where base numbers are viewed in Type 1 terms as quantitative and dimensional numbers as - relatively - in Type 2 terms as qualitative) , the Zeta 2 zeros provide the means of expressing each prime representing a dimension (in Type 2 terms) indirectly in a Type 1 manner.

In this way we are enabled to convert the ordinal members of each prime (as Type 2 qualitative) indirectly in a Type 1 (quantitative) manner.

Equally, we can convert the cardinal nature of the natural numbers as a whole (as Type 1 ) quantitative, indirectly in (a Type 2) qualitative manner through the Zeta 1 zeros.

From this perspective, the Zeta 2 can be represented as the means of conversion from the Type 2 to Type 1 aspect and the Zeta 1 as the means of conversion from Type 1 to Type 2 aspect respectively.

However when we reverse the frame of reference so that the base numbers are identified in qualitative, and the corresponding dimensional numbers in - relative - quantitative terms, these connections are reversed.

So the Zeta 2 zeros can then be represented as the means of conversion from Type 1 to Type 2 aspect and the Zeta 1 zeros as the corresponding means of conversion from Type 2 to Type 1 aspect.

Thus therefore, depending on perspective, both sets of zeta zeros play a two-way role in terms of converting between Type 1 and Type 2 (and Type 2 and Type 1) respectively.

Remember that we can use both the additive and multiplicative approaches to derive numbers!

In terms of the additive approach, each prime number can be defined as the unique sum of its natural number members (in ordinal terms).

In terms of the multiplicative approach, each natural number can then be defined as the unique product of prime number factors (in cardinal terms).

So the Zeta 2 zeros relate directly here to the additive approach and the the Zeta 1 to the multiplicative.

However ultimately all these are derived in a synchronous manner (where relationships are merely relative with everything dependent on everything else).

Thus to conclude, the Zeta 1 and Zeta 2 zeros play an equally important - and truly vital - role in enabling the seamless two-way conversion of number as between its quantitative (analytic) and qualitative (holistic) aspects.

Riemann Hypothesis (demonstration)

ReplyDeleteAlbana Diez

U. Journal of Applied Mathematic 2013.

Convergent seris.

1/3 = 1/4 + 1/18 + 1/54 + 1/162 + 1/486 + 1/1458 + 1/2187 + 1/4374

1/4 = 1/5 + 1/25 + 1/625 + 1/3125 + 1/15625 + 1/78125 + 1/312500

1/4 = 1/5 + 1/30 + 1/80 + 1/300 + 1/1440 + 1/8400 + 1/50400

1/3 = 1/4 + 1/18 + 1/54 + 1/162 + 1/486 + 1/1458 + 1/2187 + ... + ...

1/4 = 1/5 + 1/25 + 1/625 + 1/3125 + 1/15625 + 1/78125 + ... + ...