I ended the last blog entry by attempting to succinctly explain the true
significance of the famed Riemann (i.e. Zeta 1) zeros.
In fact what I said there requires just a little more clarification.
Remember the fruits of this understanding arise from a dynamic interactive manner of understanding number relationships (which always involves complementary opposite poles)!
So therefore when we start with the customary analytic view of the natural number system i.e. as independent cardinal numbers in quantitative terms, the Zeta 1 (Riemann) zeros then operate as the qualitative (holistic) counterpart of this system i.e. where the interdependence of these numbers, through their unique prime factor combinations, can be indirectly represented in a numerical fashion.
However because in dynamic terms reference frames continually switch, we can equally start with the (unrecognised) holistic view of the natural number system (where one is directly aware in an intuitive manner of the interdependence of prime factors). Then, from this perspective the Zeta 1 (Riemann) zeros operate as the quantitative (analytic) counterpart to this understanding, in providing an independent set of numbers on which the holistic understanding is necessarily grounded.
So in this interactive sense, both the natural numbers and Zeta 1 (Riemann) zeros can be seen to contain both quantitative (analytic) and qualitative (holistic) meanings, which necessarily keep switching with each other in the dynamics of experience.
And of course similar dynamics relating to continually switching reference frames likewise apply to the Zeta 2 zeros with respect to understanding of the true ordinal nature of the number system.
However it is possible to now probe more closely the exact nature of the Zeta 1 (Riemann) zeros and the clue to this again lies in the appreciation of the meaning of complementary opposite relationships.
And in this important sense, these zeros directly complement - in dynamic interactive fashion - the primes!
So from one valid perspective, we have seen that the primes and natural numbers operate in a complementary opposite fashion.
Likewise the primes as numbers without constituent factors complement those composite numbers (with factors).
Therefore in looking for the complementary opposite of the primes we should be attempting to determine all of the factors (or divisors) of the natural numbers.
So it is in this way - though the interaction of such factors - that the qualitative interdependent nature of the primes is expressed.
Now as always a lot depends on how we precisely define factors.
In conventional terms even the primes have factors with 1 being a constituent factor and the prime number itself. Thus from this perspective, each prime has 2 factors (which represents the minimum that a number can contain).
However because 1 is necessarily a factor of all numbers, just as we treat 1 as a trivial root of the number 1, we likewise treat 1 as a trivial factor of every number.
Therefore from this perspective, each prime has just 1 factor, which directly concurs with its 1-dimensional nature.
So for example if we start with 2, the independent quantitative nature of 2 as a prime is expressed through the fact that this is the only factor of 2.
Likewise with 3, the independent quantitative nature of 3 as a prime is likewise expressed through the fact that this is the only factor of 3.
However with 4 a new situation arises in that 2 and 4 are now constituent factors..
Therefore 2 now acquires a new interdependent qualitative status as a constituent factor of 4. This can equally be expressed by the fact that 2 as a prime must be now combined with another prime 2 to generate 4.
So the interdependence here arises directly through the fact that the number 2 can be expressed as part of a multiplication operation (which directly implies a qualitative transformation).
However 4 in this single context - though a composite number - acquires a relatively independent quantitative status (i.e. as a number that can be placed on the number line).
However when 4 then subsequently exists as a sub-factor of a larger number e.g. 8, then it too now acquires a qualitative interdependent status.
So each new natural number - when initially uniquely generated by prime factor combinations - carries a relative independent quantitative status. However when this number then exists as a factor of a larger number, its qualitative interdependent status is revealed.
Of course ultimately all natural number factors can be expressed as combinations of primes.
So we can say for example that for the number 8, as well as the default 8 as a factor (in a relatively independent quantitative sense), 2 and 4 are also factors (in a relatively interdependent qualitative manner).
So again though 2 is indeed a prime number, as a factor of 8 it obtains a unique qualitative status. Likewise 4 also obtains a unique qualitative status in this context (as a sub-factor of 8).
However this can also be expressed by saying that the prime combination of 2 * 2 thereby obtains a unique qualitative status (as a sub-factor of 8).
Therefore what we are saying here is that the qualitative (holistic) nature of the number system - to which the Zeta 1 (Riemann) zeros are intimately associated - directly relate to the natural factors of each member of the number system.
In other words when a number exists as a factor of another number, this implies that it is directly connected to that number through a (non-trivial) multiplication operation, and because multiplication always - when appropriately understood - entails a qualitative type transformation, this essentially therefore is what defines the qualitative (holistic) nature of such factors.
Now if we attempt to calculate the frequency of such factor combinations a surprising link exists to the harmonic series.
I have already mentioned that 2 is a factor of every 2nd number. Therefore 1/2 of all numbers contain 2 as a factor. Likewise 1/3 of all numbers contain 3 as a factor and 1/4 of all numbers contain 4 as a factor and so on.
Therefore in this way one might conclude that the average no of factors of the number n = 1/2 + 1/3 + 1/4 +...+ 1/n.
Seeing as we are leaving out 1 this would = log n – 1 + γ.
However there is a slight problem that arises with this logic in relation to discrete numbers. For example if we are counting to 10, this might suggest that 3 occurs 3 + 1/3 times, 4, 2 + 1/2 times, 6, 1 + 2/3 times, 7, 1 + 3/7 times, 8, 1 + 1/4 and 9, 1 + 1/9 times. However clearly each of these will occur just a whole number of times. Therefore to eliminate these fractions we need to make an adjustment by subtracting (1 – γ). See "Surprising Result". So this would give us log n – 2 + 2γ (or in the case where 1 is included as a factor log n – 1 + 2γ).
However if we ignore the Euler-Mascheroni constant (which arises in adjusting for discrete values) the simple formula for the average no. of factors in the number n = log n – 1.
Then the total accumulated factors to n = n(log n – 1).
This then bears a remarkable similarity with the formula for calculating the frequency of Zeta (Riemann) non-trivial zeros to t which is given as t/2π.
Thus where n = t/2π, the two formulae are identical.
Now it must be remembered that qualitative (holistic) notions of number relating to dimensions i.e. factors properly relate to the Type 2 notion of number (based on the unit circle).
And as we have seen such circular notions of number can then be converted in an imaginary linear manner (i.e. as points on an imaginary axis).
However to convert from circular to linear units we divide by 2π.
Therefore if we want to approximate the accumulated sum of factors to n, we count the frequency of non-trivial zeros on the imaginary scale to n * 2π.
So for example the accumulated sum of natural number factors to 100 will match very closely the corresponding frequency of non-trivial zeros to 628.138 (approx).
In fact what I said there requires just a little more clarification.
Remember the fruits of this understanding arise from a dynamic interactive manner of understanding number relationships (which always involves complementary opposite poles)!
So therefore when we start with the customary analytic view of the natural number system i.e. as independent cardinal numbers in quantitative terms, the Zeta 1 (Riemann) zeros then operate as the qualitative (holistic) counterpart of this system i.e. where the interdependence of these numbers, through their unique prime factor combinations, can be indirectly represented in a numerical fashion.
However because in dynamic terms reference frames continually switch, we can equally start with the (unrecognised) holistic view of the natural number system (where one is directly aware in an intuitive manner of the interdependence of prime factors). Then, from this perspective the Zeta 1 (Riemann) zeros operate as the quantitative (analytic) counterpart to this understanding, in providing an independent set of numbers on which the holistic understanding is necessarily grounded.
So in this interactive sense, both the natural numbers and Zeta 1 (Riemann) zeros can be seen to contain both quantitative (analytic) and qualitative (holistic) meanings, which necessarily keep switching with each other in the dynamics of experience.
And of course similar dynamics relating to continually switching reference frames likewise apply to the Zeta 2 zeros with respect to understanding of the true ordinal nature of the number system.
However it is possible to now probe more closely the exact nature of the Zeta 1 (Riemann) zeros and the clue to this again lies in the appreciation of the meaning of complementary opposite relationships.
And in this important sense, these zeros directly complement - in dynamic interactive fashion - the primes!
So from one valid perspective, we have seen that the primes and natural numbers operate in a complementary opposite fashion.
Likewise the primes as numbers without constituent factors complement those composite numbers (with factors).
Therefore in looking for the complementary opposite of the primes we should be attempting to determine all of the factors (or divisors) of the natural numbers.
So it is in this way - though the interaction of such factors - that the qualitative interdependent nature of the primes is expressed.
Now as always a lot depends on how we precisely define factors.
In conventional terms even the primes have factors with 1 being a constituent factor and the prime number itself. Thus from this perspective, each prime has 2 factors (which represents the minimum that a number can contain).
However because 1 is necessarily a factor of all numbers, just as we treat 1 as a trivial root of the number 1, we likewise treat 1 as a trivial factor of every number.
Therefore from this perspective, each prime has just 1 factor, which directly concurs with its 1-dimensional nature.
So for example if we start with 2, the independent quantitative nature of 2 as a prime is expressed through the fact that this is the only factor of 2.
Likewise with 3, the independent quantitative nature of 3 as a prime is likewise expressed through the fact that this is the only factor of 3.
However with 4 a new situation arises in that 2 and 4 are now constituent factors..
Therefore 2 now acquires a new interdependent qualitative status as a constituent factor of 4. This can equally be expressed by the fact that 2 as a prime must be now combined with another prime 2 to generate 4.
So the interdependence here arises directly through the fact that the number 2 can be expressed as part of a multiplication operation (which directly implies a qualitative transformation).
However 4 in this single context - though a composite number - acquires a relatively independent quantitative status (i.e. as a number that can be placed on the number line).
However when 4 then subsequently exists as a sub-factor of a larger number e.g. 8, then it too now acquires a qualitative interdependent status.
So each new natural number - when initially uniquely generated by prime factor combinations - carries a relative independent quantitative status. However when this number then exists as a factor of a larger number, its qualitative interdependent status is revealed.
Of course ultimately all natural number factors can be expressed as combinations of primes.
So we can say for example that for the number 8, as well as the default 8 as a factor (in a relatively independent quantitative sense), 2 and 4 are also factors (in a relatively interdependent qualitative manner).
So again though 2 is indeed a prime number, as a factor of 8 it obtains a unique qualitative status. Likewise 4 also obtains a unique qualitative status in this context (as a sub-factor of 8).
However this can also be expressed by saying that the prime combination of 2 * 2 thereby obtains a unique qualitative status (as a sub-factor of 8).
Therefore what we are saying here is that the qualitative (holistic) nature of the number system - to which the Zeta 1 (Riemann) zeros are intimately associated - directly relate to the natural factors of each member of the number system.
In other words when a number exists as a factor of another number, this implies that it is directly connected to that number through a (non-trivial) multiplication operation, and because multiplication always - when appropriately understood - entails a qualitative type transformation, this essentially therefore is what defines the qualitative (holistic) nature of such factors.
Now if we attempt to calculate the frequency of such factor combinations a surprising link exists to the harmonic series.
I have already mentioned that 2 is a factor of every 2nd number. Therefore 1/2 of all numbers contain 2 as a factor. Likewise 1/3 of all numbers contain 3 as a factor and 1/4 of all numbers contain 4 as a factor and so on.
Therefore in this way one might conclude that the average no of factors of the number n = 1/2 + 1/3 + 1/4 +...+ 1/n.
Seeing as we are leaving out 1 this would = log n – 1 + γ.
However there is a slight problem that arises with this logic in relation to discrete numbers. For example if we are counting to 10, this might suggest that 3 occurs 3 + 1/3 times, 4, 2 + 1/2 times, 6, 1 + 2/3 times, 7, 1 + 3/7 times, 8, 1 + 1/4 and 9, 1 + 1/9 times. However clearly each of these will occur just a whole number of times. Therefore to eliminate these fractions we need to make an adjustment by subtracting (1 – γ). See "Surprising Result". So this would give us log n – 2 + 2γ (or in the case where 1 is included as a factor log n – 1 + 2γ).
However if we ignore the Euler-Mascheroni constant (which arises in adjusting for discrete values) the simple formula for the average no. of factors in the number n = log n – 1.
Then the total accumulated factors to n = n(log n – 1).
This then bears a remarkable similarity with the formula for calculating the frequency of Zeta (Riemann) non-trivial zeros to t which is given as t/2π.
Thus where n = t/2π, the two formulae are identical.
Now it must be remembered that qualitative (holistic) notions of number relating to dimensions i.e. factors properly relate to the Type 2 notion of number (based on the unit circle).
And as we have seen such circular notions of number can then be converted in an imaginary linear manner (i.e. as points on an imaginary axis).
However to convert from circular to linear units we divide by 2π.
Therefore if we want to approximate the accumulated sum of factors to n, we count the frequency of non-trivial zeros on the imaginary scale to n * 2π.
So for example the accumulated sum of natural number factors to 100 will match very closely the corresponding frequency of non-trivial zeros to 628.138 (approx).
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