Incredible Nature of the Zeta Zeros (2)

The basic problem with the conventional approach to Mathematics is quite simple to state.

All experience - including mathematical - is of a dynamic interactive nature entailing the relative independence of distinct phenomena with their overall interdependence in holistic terms.

However as Conventional Mathematics is formally defined in terms of isolated reference frames, e.g. where objective and subjective are clearly separated, it treats mathematical objects abstractly in an independent manner.

So for example numbers (such as prime and natural) are treated in this absolute fashion as possessing an objective independent identity.

However if such numbers did indeed possess such an absolute nature, then strictly it would be impossible to recognise numbers in relation to other numbers as interdependent!


This directly implies therefore implies that such interdependent relationships can only be treated in a reduced manner. This masks therefore problems which on closer examination can be shown to be of the most fundamental nature.


Now when appropriately understood - again in dynamic interactive terms - the relative independent nature of number can be (initially) identified with its cardinal aspect in quantitative terms; the corresponding interdependent aspect, whereby numbers can be related with other numbers, can then be identified with its ordinal aspect in a qualitative manner.

Therefore from this perspective, the quantitative and qualitative aspects of number relate to cardinal and ordinal interpretation respectively.

And because the conventional approach to number - based on absolute notions of independence - is formally defined in a merely quantitative manner, this directly implies that it is not possible to deal with the corresponding ordinal aspect except in a reduced manner that distorts its very meaning.

And we will illustrate the extremely important relevance of this finding shortly!

If we start with the natural number system from the conventional (absolute) mathematical perspective these will be defined directly in a cardinal (quantitative) manner as,

1, 2, 3, 4,.....

However properly understood - in relative terms - all numbers contain two aspects which are quantitative and qualitative with respect to each other.

From this new perspective the cardinal number system is defined in terms of a default (qualitative) dimension of 1,


i.e. 1^1, 2^1, 3^1, 4^1,....


In this context I refer to the first number (that varies) as the base and the fixed invariant number as the dimension and these two numbers are quantitative and qualitative with respect to each other.


The significance of this can be easily illustrated. From the conventional perspective for example when a number is squared we concern ourselves solely in the quantitative transformation thereby involved.

So 2^2 = 4 (i.e. 4^1).

Now if we think of this in geometrical terms, a qualitative change is likewise involved whereby we move from linear (1-dimensional) to square (2-dimensional) units. So a square of 4 square units (with each side 2 units) is qualitatively distinct from a straight line that is 4 units! However from a reduced quantitative perspective this important qualitative distinction is ignored (with literally the square result expressed in 1-dimensional terms).

So when we say that 2*2 = 4, what this implies is that the reduced quantitative value of this number expression = 4. In other words we have ignored the corresponding qualitative change in the number that has thereby occurred.


Now this procedure is valid insofar as the cardinal (quantitative) aspect of number is involved (and strictly only then in a relative sense); however ultimately it leads to total confusion when we explore the corresponding ordinal (qualitative) aspect!

And this leads directly to the most fundamental issue possible with respect to number, for momentary reflection on the matter will immediately make it obvious that we cannot use cardinal notions without ordinal, or ordinal without cardinal. Put simply, we cannot attain a coherent interpretation of number without both quantitative and qualitative aspects equally incorporated in a dynamic relative manner.


So 1, 2. 3. 4 etc. in cardinal terms imply the corresponding notions of 1st, 2nd, 3rd, 4th etc. (from an ordinal perspective).


And as cardinal and ordinal notions are quantitative and qualitative with respect to each other, we therefore cannot properly formulate an interpretation of number (that is consistent) in a merely absolute quantitative manner!

Indeed not alone can we not formulate an ordinal system of number (that is coherent) in this manner, strictly we cannot even formulate a cardinal system that is consistent!



Not surprisingly, these problems lie at the very root of the problem with respect to proper recognition of the relationship between the primes and the natural numbers (and the natural numbers and the primes).


The conventional mathematical approach to looking at this relationship is unbalanced and ultimately untenable.

Approaching the issue from the (absolute) cardinal perspective, it does indeed appear that the relationship is one-way with the primes serving as the building blocks of the natural numbers (excluding 1).

So form this perspective every natural number (again other than 1) can be expressed as the unique product of prime number factors.

So for example 30 = 2*3*5 represents a unique combination of prime number factors and therefore cannot be expressed through any other combination!


However there is a key problem with this approach which is largely overlooked.

When we use a prime factor such as 5 in a cardinal sense it is taken as a single whole. However in any meaningful sense this collective whole set represents the sum of individual number objects.

So 5 = 1 + 1 + 1 + 1 + 1.

However this attempt to define 5 in cardinal terms (i.e. as the sum of individual members that are also cardinal) leads to a crucial problem.


Now clearly in common language we would readily accept that a collection of 5 necessarily includes a 1st, 2nd, 3rd, 4th and 5th member!

However when one reflects on the matter, this represents an ordinal type distinction that cannot be meaningfully derived from 5 = 1 + 1 + 1 + 1 + 1.

In other words the basis of this cardinal definition is to attempt to render number (as without qualitative distinction).

However the very notion of ordinal ranking directly implies that we can somehow distinguish each member (which thereby implies such qualitative distinction).

So, if we insist that all units are homogenous in a merely quantitative manner, then we have no means of attempting any ordinal ranking.


In other words the capacity to make ordinal distinctions comes from the holistic relationship with respect to individual members that are in some sense perceived as possessing a unique quality.

Thus the (absolute) cardinal approach where number is abstractly understood in absolute quantitative terms (as independent) cannot therefore explain the overall holistic relationship as between numbers. Therefore, it has no means within its own definitions of explaining the corresponding ordinal notion of number (without gross reductionism being involved).


And as we cannot even begin to properly deal with cardinal without equally implying ordinal notions, then it cannot provide a coherent interpretation of the cardinal aspect (again without reductionism)!


As I say these represent the most fundamental issues possible with respect to the number system.

Again putting it bluntly I have come to see clearly over the years that the present accepted approach to Mathematics is quite simply not fit for purpose.

So in the next blog entry we will come back to exploring more closely the ordinal nature of number and how it can be given a coherent qualitative interpretation.

Incredible Nature of the Zeta Zeros (1)

From the onset let us abandon any notion that the significance of the non-trivial zeros with respect to Riemann's Zeta Function can be understood in a linear rational manner.

And as their true nature greatly transcends such understanding this directly poses a dilemma for conventional mathematical interpretation (which is formally based on linear reason).

Therefore, as Hilbert rightly suggested, these zeros, when appropriately appreciated, relate to what is truly the greatest secret underlying the nature of the phenomenal - not alone the mathematical - universe, their very nature serves as an entry into the direct experience of ineffable mystery.


Though in fairness, there is now a growing appreciation of the possible physical significance of these zeros, there is little or no realisation yet of their enormous potential psychological implications.

Indeed I have little doubt that in future times the non-trivial zeros will serve an extremely important holistic scientific role in preparing spiritual aspirants to attain a pure state of meditation consistent with maintaining the highest degree of involvement in phenomenal concerns. In this way the zeros will be seen as an invaluable aid in the quest for the fullest expression of life!

Putting it bluntly the present (standard) interpretation of what constitutes Mathematics is of an extremely limited nature.
So in effect we have taken an important special case of linear (1-dimensional) understanding, where qualitative is reduced to quantitative interpretation and misleadingly elevated this as solely synonymous with valid Mathematics.

Nothing however could be further from the truth. In fact a potentially unlimited set of possible alternative interpretations of mathematical relationships exists (each one of which possesses a partial relative validity). And in all of these cases both quantitative and qualitative type aspects are related in a dynamically interactive manner.

So properly understand all mathematical activity is of a dynamic relative nature, where quantitative and qualitative aspects of understanding (which are complementary) interact.

Quite simply therefore appropriate interpretation of such activity must also formally recognise that like the blades of a scissors, Mathematics possesses two equally important aspects that are quantitative and qualitative (and qualitative and quantitative) with respect to each other.

Now, when seen from this wider dynamic perspective, there is indeed an especially important limiting case where the focus is placed primarily on quantitative meaning.
But rather like the role of Newtonian Physics, this special case should be seen as but a useful approximation with respect to mathematical reality, which inherently is of a dynamic nature and necessarily subject to uncertainty.


Indeed because of the extreme quantitative bias that defines the formal presentation of mathematical ideas, we have increasingly lost any clear notion in our society of what the qualitative aspect of Mathematics might even entail!

Though from one perspective, I can greatly admire the incredible abstract ability and indeed sheer brilliance of so many professional mathematicians, I would also see that in the main they remain blind to obvious deficiencies in the procedures that they adopt without question.


So the first requirement in approaching the nature of the non-trivial zeros is acceptance of the key fact that the number system itself must be interpreted in a dynamic interactive manner (with quantitative and qualitative aspects that are complementary).

Therefore we will first start by considering both of these aspects as relatively independent of each other.

So we have - what I term - the Type 1 aspect that is geared to the quantitative interpretation of mathematical symbols (in the standard analytical type manner using linear reason).

Then we equally have the unrecognised Type 2 aspect where the same mathematical symbols are now interpreted in a new holistic manner (using a more paradoxical circular type reasoning).


All mathematical understanding entails both reason and intuition. In brief the quantitative aspect relates to the (linear) rational and the qualitative directly to the holistic intuitive aspect respectively.

However this intuitive aspect can then be translated indirectly in a circular rational manner (using a paradoxical form of logic).

So the Type 1 aspect entails the relative specialisation in (linear) reason though intuition must also necessarily be involved.

The Type 2 aspect entails the relative specialisation of intuition, indirectly expressed in a paradoxical rational manner, though (linear) reason must also be employed.

Thus both aspects are complementary, with the fullest Type 3 expression entailing the most refined interaction of both intuitive and rational understanding.


Now the key relationship of how the primes are related to the natural numbers (and natural numbers to the primes) can be coherently constructed in isolation through both the Type 1 and Type 2 aspects. However as we shall see, they lead to interpretations - while consistent within their own frames of reference - that are in fact paradoxical in terms of each other!

So it is through the remarkable resolution of this paradox (i.e. of the two way-relationships of the primes and natural numbers) that the non-trivial zeros attain their key importance.

They in fact lead to a new distinctive Type 3 interpretation of number that is inherently of the most dynamically interactive possible, where both quantitative and qualitative aspects are reconciled.

From this new perspective, the very significance of the non-trivial zeros relates to the fundamental requirement of reconciling two modes of interpretation (that are uniquely distinct in isolated terms). Clearly when seen in this light, the true nature of the non-trivial zeros cannot be approached in a conventional mathematical manner. Not alone is this defined solely in Type 1 terms (with mere focus on quantitative interpretation) but in an absolute - rather that relative - independent fashion.


I have stated the direct consequence of this numerous times before without a hint of hyperbole. Not alone can the Riemann Hypothesis neither be proved (nor disproved) in conventional mathematical terms; its true significance cannot even be approached from this perspective.


In fact the non-trivial zeros serve as the fundamental requirement for the most advanced from of mathematical understanding (which I refer to as Type 3).

Both the Type 1 and Type 2 aspects can then be clearly seen as merely relatively independent expressions of what is interactively combined in Type 3.


Sometime in the future, Mathematics will be directly understood in Type 3 terms.
We are a long way off from that day. However the very fact that I am discussing such a development in this blog, indicates that the huge transformation process thereby required with respect to Mathematics has already started.

Remarkable Relationships (4)

Once again any root of 1 can be expressed in terms of both (real) cos and (imaginary) sin components.

Now if we consider the n roots of unity (where n is an odd values integer ≠ 1), the absolute value of the product of all the cos values = 1/{2^(n - 1)}.

The absolute value of the product of all the sin values in turn = n/{2^(n - 1)}. (In the case where the root = n/n, the sin value = o; so this value is ignored in obtaining the product of sin values!)

For example the where n = 3, the 3 roots of unity are

1
- .5 + .866i and
- .5 - .866i

Therefore the absolute value of the product of the 3 cos values = .25 = 1/(2^2)

Likewise the absolute value of the product of the 2 (significant) sin values = .75 = 3/(2^2).


However when n is even this relationship does not hold. For example when n = 2, the 2 roots are

+ 1 and
- 1

So the absolute value of the product of the cos values = 1. However no product of sin values exists in this case!

Then when n = 4, the 4 roots are

+ 1
- 1
+ i and
- i

So the absolute value of the product of (nonzero) cos values = 1 while the absolute value of (nonzero) sin values also = 1.

So in the behaviour of these values we see a pattern that is akin to the Riemann Zeta Function where the nature of the Function for odd integer values of s is uniquely distinct from corresponding even values!


As all prime numbers other than 2 are odd, this implies that where n = p (with p a prime number ≠ 2), the the absolute value of the product of of the real (cos) part of prime roots of 1 = 1/{2^(p - 1)} whereas the corresponding absolute value of the imaginary (sin) part of prime roots of 1 = p/{2^(p - 1)}


There is also an unexpected connection (with respect to the product of the real part with number partitions (where each arrangement represents a new partition).


For example in standard terms (without rearrangement) 3 has 3 partitions i.e. 3, 1 + 2 and 1 + 1 + 1. However when we allow each permutation to represent a fresh partition, then 3 has 4 partitions i.e. 3, 1 + 2, 2 + 1 and 1 + 1 + 1.

So the number of partitions here of 3 (allowing rearrangement) = 2^2.

More generally the number of partitions of n (allowing rearrangement) = 2^(n - 1).

Where n is a prime number (= p) the number of partitions (allowing rearrangement) = 2^(p - 1).

And this result is the inverse of the absolute product of the real part of the prime roots of 1.

So in this way we can perhaps see an intimate connection as between partitions and the prime roots of 1!

We have seen before that whereas the cardinal nature of number corresponds to linear logic, the ordinal nature (pertaining to the relationship between numbers) strictly corresponds to an alternative circular logic.

Therefore just as the roots of 1 represent the most basic example of such linear/circular behaviour, not surprisingly partitions (pertaining to a fundamental relationship between numbers) are characterised by the same linear/circular behaviour.

Remarkable Relationships (3)

In an earlier posting "Interesting Prime Result", I showed how the (finite) Zeta 2 equation could be used to detect whether a number is prime.

So once the Zeta 2 Equation is defined as:

s^1 + s^2 + s^3 + s^4 +.......s^n = 0

Then dividing by the trivial solution i.e. s = 0 we obtain

1 + s^1 + s^2 + ....... s^(n - 1) = 0.


Nest by letting s = 1 in the expression


y = 1 + s^1 + s^2 + ....... s^(n - 1), by a process of continued differentiation (with respect to s) we showed how to determine whether a number is prime.


However we could also seek to proceed in a complementary direction through obtaining the integral of the same simple expression i.e.

y = 1 + s^1 + s^2 + ....... s^(n - 1).


So ʃy ds = s + (s^2)/2 + (s^3)/3 + (s^4)/4 + .... + (s^n)/n.


Then setting s = 1, we obtain the first n terms in the harmonic series i.e.


1 + 1/2 + 1/3 + 1/4 +..... + 1/n.

Now as n becomes very large the sum of this series approximates very close to log n (which measures the average spread as between prime numbers in the region of n).

For example when n = 1,000,000 the sum of the first n terms = 14.384.

So this provides an approximate measurement of the average spread (or gap) as between prime numbers (in the region of 1,000,000).
Once again this approximation will steadily improve (in relative terms) as n increases.

Thus it is interesting how a simple process of differentiation on this simple (Zeta 2) expression can determine on the one hand whether a number is prime, while the corresponding process of integration can establish the nature of the general distribution among the primes!


If we just concentrate on the first terms of the simple expression we get 1.

Then if we successively integrate with respect to s we obtain s/1!, s^2/2!, s^3/3!,
s^4/4! and so on.

Therefore by adding all these terms we obtain a simple formula for e^s (containing the first n terms of the corresponding infinite expression for e^s).

Of course where s = 1, we approximate the value of e, which becomes ever more accurate as n increases.



Returning to the roots of 1 we showed again how the average value of these roots where n is prime approximates 4/π (especially with respect to the arithmetic mean).

And this approximation steadily improves as the value of n increases.

Once again it is important to bear in mind that we use a reduced linear (1-dimensional) quantitative approach in calculating such values. This means that negative values are treated as positive and imaginary values are treated as real!

However it would also be possible to calculate values using a reduced 2-dimensional quantitative approach.

This entails in effect that negative (as well as positive) values are now considered with however imaginary once again converted to real format.


Now if we attempt to obtain the sum of roots using this approach negative will exactly cancel positive values with result = 0.

However if we obtain the product of such roots a non-trivial result will emerge which in all cases (where n is odd) can be expressed as 1/{(2^(n - 1)/2} in absolute terms.


For example where n = 5, the five roots of 1 (expressed in this 2-dimensional manner) are (to 9 decimal places),


.309016994 + .951056516 = 1.260073510
- .809016994 + .587785252 = - .221231742
.309016994 - .951056516 = - .642039522
- .809016994 - .587785252 = - 1.396802246
1 + 0 = 1


So when we multiply these 5 roots we obtain - .25.
And as in this case (n - 1)/2 = (5 - 1)/2 = 2, the absolute value of .25 = 1/(2^2) conforms to the general formula.

Though the answer in this case is negative, it can also be positive (as for example where n = 9)

Remarkable Relationships (2)

It struck me after I completed the last entry that it was attempting to equate two different types of mean average i.e. the arithmetic and geometric respectively.

Now it is well known that in the non-trivial case where absolute values of numbers differ that the arithmetic mean will always be greater in magnitude that the geometric.

However where the values are all fairly close to 1, the difference in magnitude will be quite small.

For example if we take the five numbers 1, 1.1, 1.2, 1.3 and 1.4 the arithmetic mean = 1.2 and the geometric mean = 1.191596...

Now when we look at the absolute value of the n roots of 1, the numerical value necessarily lies as between 1 and 1.4142... (i.e. the square root of 2).


So we would expect in this case the geometric mean for the n roots to be near to the arithmetic mean (though necessarily smaller).

However what is interesting in the case of the absolute value of these these n roots is that both the arithmetic and geometric means continue to converge more closely on the limiting value 4/π (with however the arithmetic converging more rapidly to this value).


For example when n = 17, the geometric mean = 1.2657221 (with 4/π = 1.27323954...). However when n = 31 the geometric mean = 1.266131209 (which is closer to the limiting value). As we would expect the arithmetic mean for n = 31 is much closer to 4/π (i.e. 1.2729671...).


This convergence towards the limiting value of 4/π, also appears to apply when we confine ourselves merely to the prime numbered roots.


So for example once again when n = 31, there are 11 prime numbered roots i.e. the 2nd, 3rd, 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 29th and 31st respectively.

Now the arithmetic means of these 11 roots = 1.2710048... (which again is very close to 4/π = 1.27323954...). The geometric mean of these same 11 roots = 1.26216 which again reasonably approximates the limiting value).


In fact this highlights a key feature of the linear/circular nature of prime numbers . In other words we can make the important observation that it is in the very nature of prime number behaviour that the average absolute value (both arithmetic and geometric) of all n roots - when n is very large - approximates ever more closely the corresponding arithmetic and geometric averages (entailing the prime numbered roots only).


Thus though though we are concentrating here on the reduced absolute interpretation of root values, they are actually expressive of the linear/ circular behaviour of the prime numbers (with respect to the natural numbers) which is inherent to their very nature.


One remarkable final illustration of this can be given. We have seen repeatedly the importance of 2/π (and counterpart 4/π) in explaining the (reduced) quantitative behaviour of prime numbers (where negative values are treated as positive and imaginary values as real).


In quantitative terms 2/π = i/log i.

Now if we again attempt the same reduced (imaginary to real) quantitative transformation on this relationship we obtain 1/log 1 i.e. 1/0.

Now from a quantitative perspective, this result is somewhat meaningless. However if we now switch back to qualitative interpretation 1/0 implies the relationship as between linear and circular meaning (i.e. quantitative and qualitative). And this precisely defines the relationship as between the prime and natural numbers (in qualitative terms)!

Remarkable Relationships

I have long commented on the linear (1-dimensional) nature of Conventional Mathematics from a qualitative perspective. This implies a merely uni-polar approach i.e. where objective is clearly separated from subjective, quantitative from qualitative etc.

Put another way it implies that mathematical meaning is merely posited (in a real conscious rational manner).

Thus though negative and imaginary operations are certainly possible within Conventional Mathematics in quantitative terms, these are always strictly interpreted in a linear (1-dimensional) manner from a qualitative perspective.

Furthermore, as I have repeatedly pointed out, the conventional understanding of both the Prime Number Theorem and the Riemann Hypothesis takes place within this limited linear perspective.


As we know the circular roots of unity (apart from the 1st and 2nd dimensions) entail complex numbers with positive and negative values. However though these roots inherently entail a dynamic relationship as between linear and circular notions, in conventional mathematical terms they are interpreted in a merely reduced linear manner (i.e. where the qualitative aspect is directly reduced in quantitative terms).


However we can invert this whole approach whereby now mathematical operations are considered quantitatively in a real positive manner, with qualitative interpretation taking place in a multi-dimensional fashion.

What this entails is that we simply interpret with respect to quantitative operations both negative and imaginary values in a positive real fashion.

For example the 3 roots of 1 are 1, -.5 +.866i, and -.5 - .866i (correct to 3 decimal places).

Now interpreted in a real positive quantitative manner these are 1, .5 + .866 and .5 + .866 i.e. 1, 1.366 and 1.366.


Now if we obtain the mean average of these 3 roots, i.e. (1 + 1.366 + 1.366)/3 = 1.244.

This is already a reasonable approximation to the value of 4/π = 1.27323954...
and this approximation quickly increases as the number of roots of 1 also increases.

So confining ourselves to prime numbered roots, the mean average of the 17 roots of 1(in this real positive manner) = 1.2723335... which is already very close to 4/π =
1.27323954...


Thus as we increase the number of prime numbered roots of 1, the mean average value approximates very quickly to 4/π.

Now the significance of 4/π is that it serves as a perfect archetype of linear to circular meaning representing both the length of the perimeter of the square to its inscribed circular circumference and equally the area of the same square to the area of its inscribed circle.








Now if we take the same n roots of 1 and then multiply them (in this real positive manner) before then raising to the power of 1/n once again the result will approximate to 4/π.

For example the product of the 3 roots of 1 i.e. 1 * 1.366 * 1.366 = 1.865956 and when we raise this to 1/3 we obtain 1.23112... which is already a reasonable approximation to 4/π = 1.27323954...

And this approximation to 4/π again improves (though more slowly than in the previous case) as we increase the number of roots of 1.

So again using 17 roots to illustrate, when we multiply these together and then raise to 1/17 we obtain 1.2657221... (which is much closer to 4/π).

Thus there is a remarkable link quantitatively here as between addition and multiplication on the one hand and multiplication and exponentiation on the other.


Therefore with respect to the n roots of a given prime number (taken in a reduced real positive manner),

(1st + 2nd + 3rd ... + nth)*(1/n) → (1st * 2nd * 3rd ... * nth)^(1/n) → 4/π with the approximation improving as n gets larger. So in the limit as n → ∞, equality as between the three expressions is achieved (in a relative manner).


In the past I have continually expressed the viewpoint that that nature of prime number behaviour cannot be properly understood without the incorporation of both linear (analytic) and circular (holistic) modes of qualitative interpretation.


Now here in a reverse quantitative manner we can see the same perfect relationship as between linear and circular notions with respect to number behaviour.

So once again this clearly demonstrates that it is not enough for example to attempt to define linear and circular notions in a merely quantitative manner (as in Type 1 Mathematics).

Equally they must be defined in a qualitative manner (Type 2 Mathematics).

And then comprehensive understanding can only materialise through the synchronised interaction of both quantitative and qualitative meaning (Type 3 Mathematics).

Prime Number Theorem: Alternative Formulation

In a short addendum (on the "The Euler Identity" blog), a simple approximation for log n (with n positive)was provided i.e.

log n → (n^x - 1)/x as x → 0.


The Prime Number Theorem (providing the general distribution of the primes among the natural numbers) in turn can be simply expressed by the expression n/log n (as n → ∞).

Therefore substituting our approximation for log n, the Prime Number Theorem could be expressed as nx/(n^x - 1) as n → ∞ and x → 0!


It is often expressed that the secret of prime number behaviour lies in the relationship as between addition and multiplication!

It could equally be said that this secret lies in the relationship as between multiplication and exponentiation (which is simply demonstrated by this expression).

Also we have the combination of two extremes whereby the relative approximation is continually improved through making one variable (n) progressively larger, while the other variable (x) is made progressively smaller.

This likewise captures the essence of prime number behaviour which represents an extreme as between linear (quantitative) and circular (qualitative) aspects, whereby the relative independence of each prime number (as discrete) is made compatible with the overall interdependence of prime numbers (as continuous).


We also had provided a complementary Prime Number Theorem where the average absolute value of prime numbered roots of 1 approximates to 2/π = i/log i.


Once again i/log i can be approximated as ix/(i^x - 1) as x → 0.

So 2/π → ix/(i^x - 1) as x → 0.