It would be instructive once again to attempt to convey the dynamic meaning of the Zeta 1 (Riemann) zeros.
And it this respect one can obtain assistance from looking at the corresponding nature of the Zeta 2 zeros (especially with respect to the simplest 2-dimensional case).
So the 2 roots of 1, serving as the indirect quantitative expression of the qualitative notions of 1st and 2nd (in the context of 2 members) are + 1 and – 1 respectively.
Now the relative interdependence of these two members, arises through the fact that the sum of roots = 0. This reflects that ordinal positions are relative and can arbitrarily switch depending on context.
However the relative independence is expressed through each individual root (i.e. + 1 and – 1 respectively).
So the key point here is that a perfect dynamic balance is maintained as between independent and interdependent aspects (in this interactive context).
I have explained this many times in relation to our understanding of left and right turns at a crossroads.
When one approached the crossroads for example heading in a N direction, left and right turns have a relative independent existence in an unambiguous manner. So if we designate a left turn in this context as + 1, then a right turn is thereby – 1 (i.e. not a left turn).
Again when approaching the crossroads heading in a S direction, left and right turns again have a relative independent existence, in an unambiguous manner. So if again we designate a left turn in this alternative context as + 1, then a right turn is – 1 (i.e. not a left turn).
However when we now envisage approaching the crossroads from both N and S directions (simultaneously) then left and right have a relatively interdependent existence (in a paradoxical fashion).
So what is left (i.e. + 1) when approached in a N direction, is right (i.e. – 1) when approached in a S direction; and what is right (i.e. – 1) when approached in a N directions is left (i.e. + 1) when approached is a corresponding S direction.
Thus we are able to combine relative relative independence (in the interpretation of turns when approached from just one direction with relative interdependence (in the interpretation of turns simultaneously from both directions).
Now the former unambiguous understanding of relative independence (where the designation of number remains fixed) corresponds to rational understanding of a quantitative (analytic) kind.
However the latter understanding of relative interdependence (where the designation of number switches) corresponds to intuitive understanding of a qualitative (holistic) nature which then indirectly can be given a rational interpretation in a (circular) paradoxical fashion.
Now these two types of understanding - which implicitly are involved in the commonplace understanding of two turns at a crossroads - are utterly distinct from each other.
The quantitative (analytic) aspect - relating to relative independence - corresponds to the default root of the 2 roots of 1 (=+ 1); the qualitative (holistic) aspect relating to relative interdependence - corresponds to other root (= – 1). In dynamic interactive terms this implies paradox in that the sign of 1 continually switches depending on context.
However the qualitative (holistic) aspect is completely reduced in conventional mathematical interpretation (and identified in a merely quantitative manner).
Furthermore the quantitative aspect is identified in an absolute manner, though from an appropriate dynamic interactive perspective it is strictly relative.
Though perhaps somewhat more difficult to appreciate, the Zeta 1 (Riemann) zeros, likewise operate in a dynamic complementary manner.
However here - instead of focussing on the internal ordinal members of a number group - we concentrate on the collective relationship of the primes to the natural numbers in cardinal terms.
Now just as our approach to the crossroads was taken from two directions, equally we can look at this relationship as between the primes and natural numbers from two perspectives.
From the standard quantitative, perspective we can attempt to measure for example the frequency of prime numbers up to a given natural number.
How from an alternative quantitative perspective we could attempt to measure the frequency of all natural number to (distinct) prime factors.
In both cases one will obtain relatively independent unambiguous answers in a quantitative manner.
So one common approximation for the first case is given by n/log n.
And a corresponding approximation in the second case is given as log n/loglog n.
Then if we denote log n as n1, then this second approximation is given as n1/log n1.
However - as with our approach to the crossroads from two opposite directions - if we now attempt to view simultaneously the two relationships as between the primes and natural numbers, i.e. in a relatively interdependent manner, then inevitable paradox is involved.
Thus from the standard perspective the relationship appears to be between primes and natural numbers (considered as base numbers). However from the alternative perspective, the relationship appears to be between natural numbers and primes (considered as representing the dimensional aspect of number as factors).
So in fact there are two complementary relationships entailing the relationship as between primes and natural numbers.
Thus when we properly view this twin relationship in a dynamic interactive manner, complete paradox results. From one perspective, the relationship is between primes and natural numbers (in Type 1 terms) . From the alternative perspective the relationship is between natural number and prime factors (in Type 2 terms).
Therefore from the holistic (qualitative) perspective - which simultaneously combines both complementary reference frames - one realises that the primes and natural numbers are fully interdependent with each other (ultimately in an ineffable manner).
From the analytic (quantitative) perspective - taken from just one partial perspective - one realises that both primes and natural numbers are relatively independent of each other (both as base numbers and dimensional factors) .
Now just as the roots of 1 - as the indirect quantitative expression of ordinal notions - play this role of balancing notions of quantitative independence and qualitative interdependence, the non-trivial zeros play the same role with respect to the primes in cardinal terms.
Thus when we start by viewing the primes and natural numbers in a relatively independent quantitative manner, each Zeta 1 zero expresses the holistic qualitative nature of this relationship as a point (on an imaginary line through .5) where both of the perspectives we have looked at are simultaneously valid. What this entails is that at each point there is no distinction as between the nature of a prime or natural number (or alternatively the notion of a base of dimensional number).
This however has no direct meaning in a linear rational manner but rather relates directly to pure intuitive understanding (as a psycho-spiritual energy state).
So the simplest way of explaining each trivial zero is as a pure energy state of pure number interdependence (with complementary physical and psychological interpretations) where the cardinal notion of number (as representing a fixed form) loses any residual meaning.
Now each zero in a sense still has a certain form as a complex number with a transcendental imaginary part. However this represents the most elusive and highly dynamic nature of number possible, as the final bridge as between form and ineffable emptiness.
So we always can only hope to approximate to such ultimate understanding in a dynamic experiential manner.
However the trivial zeros, in their collective identity, equally have a relatively independent status (in a quantitative manner) . And it is this aspect that was so wonderfully exploited by Riemann to zone in precisely on the measurement of the primes up to any given number.
And it this respect one can obtain assistance from looking at the corresponding nature of the Zeta 2 zeros (especially with respect to the simplest 2-dimensional case).
So the 2 roots of 1, serving as the indirect quantitative expression of the qualitative notions of 1st and 2nd (in the context of 2 members) are + 1 and – 1 respectively.
Now the relative interdependence of these two members, arises through the fact that the sum of roots = 0. This reflects that ordinal positions are relative and can arbitrarily switch depending on context.
However the relative independence is expressed through each individual root (i.e. + 1 and – 1 respectively).
So the key point here is that a perfect dynamic balance is maintained as between independent and interdependent aspects (in this interactive context).
I have explained this many times in relation to our understanding of left and right turns at a crossroads.
When one approached the crossroads for example heading in a N direction, left and right turns have a relative independent existence in an unambiguous manner. So if we designate a left turn in this context as + 1, then a right turn is thereby – 1 (i.e. not a left turn).
Again when approaching the crossroads heading in a S direction, left and right turns again have a relative independent existence, in an unambiguous manner. So if again we designate a left turn in this alternative context as + 1, then a right turn is – 1 (i.e. not a left turn).
However when we now envisage approaching the crossroads from both N and S directions (simultaneously) then left and right have a relatively interdependent existence (in a paradoxical fashion).
So what is left (i.e. + 1) when approached in a N direction, is right (i.e. – 1) when approached in a S direction; and what is right (i.e. – 1) when approached in a N directions is left (i.e. + 1) when approached is a corresponding S direction.
Thus we are able to combine relative relative independence (in the interpretation of turns when approached from just one direction with relative interdependence (in the interpretation of turns simultaneously from both directions).
Now the former unambiguous understanding of relative independence (where the designation of number remains fixed) corresponds to rational understanding of a quantitative (analytic) kind.
However the latter understanding of relative interdependence (where the designation of number switches) corresponds to intuitive understanding of a qualitative (holistic) nature which then indirectly can be given a rational interpretation in a (circular) paradoxical fashion.
Now these two types of understanding - which implicitly are involved in the commonplace understanding of two turns at a crossroads - are utterly distinct from each other.
The quantitative (analytic) aspect - relating to relative independence - corresponds to the default root of the 2 roots of 1 (=
However the qualitative (holistic) aspect is completely reduced in conventional mathematical interpretation (and identified in a merely quantitative manner).
Furthermore the quantitative aspect is identified in an absolute manner, though from an appropriate dynamic interactive perspective it is strictly relative.
Though perhaps somewhat more difficult to appreciate, the Zeta 1 (Riemann) zeros, likewise operate in a dynamic complementary manner.
However here - instead of focussing on the internal ordinal members of a number group - we concentrate on the collective relationship of the primes to the natural numbers in cardinal terms.
Now just as our approach to the crossroads was taken from two directions, equally we can look at this relationship as between the primes and natural numbers from two perspectives.
From the standard quantitative, perspective we can attempt to measure for example the frequency of prime numbers up to a given natural number.
How from an alternative quantitative perspective we could attempt to measure the frequency of all natural number to (distinct) prime factors.
In both cases one will obtain relatively independent unambiguous answers in a quantitative manner.
So one common approximation for the first case is given by n/log n.
And a corresponding approximation in the second case is given as log n/loglog n.
Then if we denote log n as n1, then this second approximation is given as n1/log n1.
However - as with our approach to the crossroads from two opposite directions - if we now attempt to view simultaneously the two relationships as between the primes and natural numbers, i.e. in a relatively interdependent manner, then inevitable paradox is involved.
Thus from the standard perspective the relationship appears to be between primes and natural numbers (considered as base numbers). However from the alternative perspective, the relationship appears to be between natural numbers and primes (considered as representing the dimensional aspect of number as factors).
So in fact there are two complementary relationships entailing the relationship as between primes and natural numbers.
Thus when we properly view this twin relationship in a dynamic interactive manner, complete paradox results. From one perspective, the relationship is between primes and natural numbers (in Type 1 terms) . From the alternative perspective the relationship is between natural number and prime factors (in Type 2 terms).
Therefore from the holistic (qualitative) perspective - which simultaneously combines both complementary reference frames - one realises that the primes and natural numbers are fully interdependent with each other (ultimately in an ineffable manner).
From the analytic (quantitative) perspective - taken from just one partial perspective - one realises that both primes and natural numbers are relatively independent of each other (both as base numbers and dimensional factors) .
Now just as the roots of 1 - as the indirect quantitative expression of ordinal notions - play this role of balancing notions of quantitative independence and qualitative interdependence, the non-trivial zeros play the same role with respect to the primes in cardinal terms.
Thus when we start by viewing the primes and natural numbers in a relatively independent quantitative manner, each Zeta 1 zero expresses the holistic qualitative nature of this relationship as a point (on an imaginary line through .5) where both of the perspectives we have looked at are simultaneously valid. What this entails is that at each point there is no distinction as between the nature of a prime or natural number (or alternatively the notion of a base of dimensional number).
This however has no direct meaning in a linear rational manner but rather relates directly to pure intuitive understanding (as a psycho-spiritual energy state).
So the simplest way of explaining each trivial zero is as a pure energy state of pure number interdependence (with complementary physical and psychological interpretations) where the cardinal notion of number (as representing a fixed form) loses any residual meaning.
Now each zero in a sense still has a certain form as a complex number with a transcendental imaginary part. However this represents the most elusive and highly dynamic nature of number possible, as the final bridge as between form and ineffable emptiness.
So we always can only hope to approximate to such ultimate understanding in a dynamic experiential manner.
However the trivial zeros, in their collective identity, equally have a relatively independent status (in a quantitative manner) . And it is this aspect that was so wonderfully exploited by Riemann to zone in precisely on the measurement of the primes up to any given number.
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