ζ(s) = 1– s + 2– s + 3– s + 4– s + …, how the sum over the positive integers is connected to a product over the primes for any real exponent s (where s > 1).
Then Riemann showed how this function could be extended to the complex plane for all values of s ≠ 1, thus enabling a much greater knowledge of the quantitative distribution of the primes.
However Riemann’s zeta function is interpreted in a merely analytic (quantitative) manner based on the absolute interpretation of number.
So the obvious further extension to this approach is to now view the function in a dynamic relative manner, where both analytic (quantitative) and holistic (qualitative) aspects with respect to number, interact in complementary fashion.
And in doing this, the very nature of the Riemann zeta function (and its associated Riemann Hypothesis) is utterly transformed.
My intention here is not to attempt to elucidate in detail on the holistic interpretation of mathematical symbols (which requires a radically distinctive mind-set to corresponding analytic interpretation).
However some brief illustrations are perhaps required in order to give an insight into the holistic nature of the complex plane.
I have already briefly indicated how number (especially with reference to “2”) can be given an alternative holistic interpretation (where the mutual interdependence of its ordinal units is explicitly recognised).
As all mathematical symbols can be given distinctive analytic and holistic interpretations, this therefore equally applies to positive and negative signs.
In holistic terms, + implies the positing of phenomenal meaning in a conscious manner (where objects are thereby viewed as independent of the interpreting mind)
Indeed it is such positing that creates the impression - which is strictly an illusion - that numbers possess an absolute unchanging identity.
So from a holistic perspective, all mathematical symbols are conventionally treated - in formal terms - in a solely positive manner.
By contrast in holistic terms, the – sign implies the negation of consciously posited phenomena.
This in fact is intimately related to how we truly experience number.
One starts with the positive direction, consciously relating in rational terms, for example, to the experience of number as an (external) object phenomenon.
So in order to switch to the corresponding experience of number as a - relatively - (internal) mental perception, one must negate to a degree the (external) object (strictly, undue identification with the object).
Thus the continual interaction of a number in conscious terms as both an object phenomenon and mental perception - which are, relatively, external and internal with respect to each other in experience - requires the dynamic negation of either pole (as absolutely independent). And the intuitive knowledge of the consequent interdependence of both polarities of number relates directly to the holistic unconscious aspect of understanding.
Therefore the great fallacy that underlies conventional interpretation is the treatment of number objects in an absolute independent manner.
Strictly speaking, number objects - indeed all mathematical objects - are meaningless in the absence of the mental constructs used for their interpretation.
And once we admit the twin interaction of objects and mental constructs, the very nature of number is understood as relative, with complementary opposite poles (that equally can be posited in relative independent terms). However the very switching between opposite poles - representing their corresponding interdependence in relative terms - requires the dynamic negation of either pole as separate.
And this in brief represents the holistic mathematical nature of the negative sign (–) relating directly to the unconscious aspect of mathematical understanding.
So number can be given positive (+) and negative (–) interpretations in standard analytic fashion; equally number can be given positive (+) and negative (–) interpretations in an (unrecognised) holistic manner.
Now one might be initially tempted to dismiss all this as somewhat irrelevant to mathematics as we know it. However this would be very mistaken.
Let me simply illustrate how this holistic interpretation of the negative sign (–) is profoundly important to mathematics.
If we start with a simple integer such as 3, we can more fully represent this as 31. So the dimensional aspect is defined in positive terms.
However, when we now change the dimensional number of 1 to its negative sign, we get
3– 1 = 1/3. So we switch from the whole notion of 3 to the corresponding part notion of 1/3, through switching with respect to the dimension from + 1 to – 1. And if we now represent 1/3 in the standard analytic manner as (1/3)1, then to switch back to whole recognition of 3, once again we must negate the dimension of 1.
The importance of this illustration is to show that the very means, by which one is enabled to switch from whole to part recognition (and in turn from part to whole recognition) in mathematical terms, is through the intervention of the unconscious.
Because in holistic terms, conventional interpretation is merely confined to the conscious positive direction (+ 1) this invariably leads to the reduction of the whole notion as the quantitative sum of its various parts.
And this is vital in turn to appreciate a fundamental problem with respect to interpretation of the Riemann zeta function.
We have seen that when s > 1, function values intuitively correspond to conventional analytic notions of the sum of a series (where the whole is represented as the sum of its various parts).
However, with the Riemann zeta function, values for 1 – s (as summed in an analytic manner) no longer correspond intuitively with actual results.
The reason for this is that we must switch from part to whole type appreciation of the sum of a series, where it now represents the unique interdependence of various sub-wholes (not directly related to separate parts).
And this switch in the manner of understanding from part to whole type recognition as one moves from function values (s > 1) for ζ(s) to corresponding function values for ζ(1 – s) requires a corresponding switch from the conscious (positive) to the unconscious (negative) aspect of understanding.
This negative unconscious direction (with respect to linear appreciation), is especially revealing in the case of primes.
To illustrate, this let us take the prime number 7, which can be more fully represented (in Type 1 terms) as 71.
When we now express this whole number with respect to its negative dimension, – 1,
we obtain 7 – 1 = 1/7.
However the crucial holistic significance of this behaviour is completely missed from the conventional perspective.
So we start with the number 7 i.e. 71 as a prime, considered as a “building block” of the natural number system in 1-dimensional terms. In this sense (because 7 contains no sub-factors) it represents an example of the most independent type of number from a linear perspective.
But when we obtain the reciprocal of 7, i.e. 7 – 1, we now obtain an example of the most circular of numbers (i.e. a cyclic prime) where a high level of interdependence attaches to its unique 6-digital sequence. For example, when we multiply 142857 by 1, 2, 3, 4, 5 and 6 respectively, we derive the same circular sequence of digits.
The holistic significance of this could not be more important, as it implies that from the unconscious perspective, prime numbers are revealed as the most circular i.e. interdependent of all numbers.
And we have already explained this interdependence, in terms of the special qualitative uniqueness of its ordinal members, which indirectly can be expressed in quantitative terms as equidistant points on the circle of unit radius (in the complex plane).
Thus again as a “building block”, a prime appears as the most independent of all numbers in a quantitative (analytic) fashion; however, considered as a group comprising its ordinal natural number members, a prime by contrast appears as the most interdependent of all numbers in a qualitative (holistic) manner.
And these two complementary aspects of the primes - quantitative and qualitative - require explicit interpretation with respect to both the conscious (+) and unconscious (–) aspects of understanding, in a dynamic interactive manner.
In conventional analytic terms, i relates to the square root of – 1. So when x2 = – 1,
x = i (strictly + i or – i). So what we have here is an attempt to express – 1, which relates to a 2-dimensional definition, in a reduced 1-dimensional manner.
In holistic terms – 1 (as we have seen) represents the unconscious aspect of the understanding of 1. However this causes a switch to 2-dimensional understanding, where phenomena are now understood as having two polar opposite meanings, which are dynamically interdependent with each other. Thus in a dynamic sense, negation necessarily implies an - already posited - object in an intuitive fusion of psycho-spiritual energy.
So in this context, i represents the indirect attempt to convey holistic meaning in a reduced linear rational manner. And this holistic interpretation of the imaginary notion is precisely what I have been illustrating throughout this contribution!
Thus whereas the standard analytic, represents - in qualitative terms - the real aspect of mathematical understanding, the (unrecognised) holistic aspect, indirectly conveyed in a linear rational manner, represents the corresponding imaginary aspect.
So, just as numbers can be given real and imaginary interpretations in an analytic (quantitative) manner, likewise, numbers can be given real and imaginary interpretations in a corresponding holistic (qualitative) fashion.
And a comprehensive interpretation of the Riemann zeta function (which is defined with respect to the complex plane) thereby requires both analytic and holistic interpretation in a dynamic interactive manner i.e. where numbers can have a real or imaginary meaning in both quantitative and qualitative terms.
I will just briefly attempt here to indicate the enormous significance of the holistic nature of the complex plane.
From an analytic (quantitative) perspective, the one value for which the Riemann zeta function is undefined is where s (representing an exponent, i.e. dimensional number) = 1.
It is quite similar from a holistic (qualitative) perspective, where again the one value for which the Riemann zeta function (Zeta 1 function) remains undefined is where s = 1.
And it is precisely the same for the infinite version of the Zeta 2 function (which is dynamically interdependent with the Zeta 1).
So 1 + x1 + x2 + x3 + … = 1/(1 – x)
Thus when x = 1, we obtain an infinite value, which means that the function is not defined for this value.
However the holistic interpretation is very revealing, for 1 in this context represents 1-dimensional i.e. linear interpretation, which defines the conventional mathematical approach!
Therefore, it clearly entails that from this enlarged dynamic perspective - and it is necessary to pause here to fully grasp the significance of what follows - the Riemann zeta function (and associated Zeta 2 function) cannot be properly defined in the conventional mathematical manner, where number is considered as absolute.
Thus the true nature of number is relative in a dynamic interactive fashion, with conventional mathematics representing but a special - though necessarily limited - case with interpretation taking place in 1-dimensional terms (i.e. where the qualitative aspect of relationships is reduced in a quantitative manner). Here a solely analytic (quantitative) interpretation of number is possible.
However for all other dimensional numbers (≠ 1), both analytic and holistic interpretations of number exist in a relative manner.
And from a dynamic interactive perspective, the Riemann zeta function is only properly defined for all these relative interpretations!
One remarkable example of this relative complementarity - as already mentioned - is provided through Riemann’s important functional equation, where for any value of the function ζ(s), for a positive s (> 1) on the real right hand axis, a corresponding value of the function can be given for 1 – s on the left hand axis.
So for example, if we know the value for s = 2 i.e. ζ(2), we can then, through Riemann’s equation, calculate the corresponding value for ζ(– 1).
And is well known, the value for, ζ(2) = 1 + 1/22 + 1/32 + 1/42 + … = π2/6 = 1.6449…
Now what is interesting about this value that it represents in quantitative terms, an irrational (transcendental) number with a standard rational interpretation (from a qualitative perspective).
So, the convergent value of the expression conforms to the standard rational interpretation of the value of such a series.
And in such standard (1-dimensional) interpretation, each successive term is treated (in a reduced manner) as a point on the real number line. Then the total value of the series is understood as the sum of successive part values (for each term).
Therefore, the whole is expressed in a reduced manner as the sum of its successive parts.
However though each term represents a rational number, the sum of terms is irrational (i.e. transcendental). Thus, clearly there is an unexplained qualitative dimension to the very form of a transcendental number (that reflects the combination of finite with infinite notions).
So once more, ζ(2) has an irrational (transcendental) value in quantitative terms, with a standard rational interpretation from a qualitative perspective.
Now, the key to properly appreciating the corresponding value of ζ(1 – 2), i.e. ζ(– 1) is that the two values i.e. ζ(2) and ζ(– 1) are linked in a dynamic complementary manner.
Thus, whereas the value of ζ(2) i.e. π2/6, represents an irrational (transcendental), by contrast the value of ζ(– 1) = – 1/12, represents a rational value in quantitative terms.
However ζ(– 1) = 1 + 2 + 3 + 4 + …, which from the standard rational perspective = ∞.
So clearly, we cannot interpret its result (i.e. – 1/12) with respect to the Riemann zeta function in the accepted rational manner.
Rather, because of the complementary nature of results connected through Riemann’s function, the interpretation we should now give to ζ(– 1) = – 1/12, is of an irrational (transcendental) form in qualitative terms!
We can get a deep insight into what this means through first looking at the nature of gπ from an analytic perspective, which represents the important relationship as between the circular circumference and its line diameter.
In a corresponding holistic manner, π represents an important relationship as between circular and linear type understanding (i.e. where both are in a necessary balance with each other)
Therefore one has to abandon here the linear intuition that the total quantitative value of a series represents the reduced sum of its separate (independent) terms, which again conforms to standard rational type interpretation.
Rather we must now recognise that in summing a series, both analytic notions of (linear) independence and holistic notions of (circular) interdependence are mutually involved.
To illustrate this, it is perhaps better to start with the slightly simpler value (according to the Riemann zeta function) for ζ(0) = – 1/2.
ζ(0) = 1 + 1 + 1 + …, which from the conventional analytic perspective leads to an infinite result for the series.
Therefore the obtained result i.e. – 1/2 appears non-intuitive in conventional rational terms.
However we can make progress by looking initially at the alternating eta series, i.e.
η(0) = 1 – 1 + 1 – 1 + …
In summing this series, if we take an even number of terms (where terms are naturally arranged in circular fashion as complementary couplets) the result = 0.
However if we take an odd number of terms (where each is treated as independent in a linear manner) the result = 1.
And accepting an equal probability that the series contains an even or odd number of terms respectively, we could therefore express the sum of this eta series as the average of the two possible results (1 and 0) = 1/2.
However in doing this, we thereby abandon a strict linear approach, to obtain an answer that involves in effect a balanced compromise as between linear and circular notions.
Then a simple transformation in linear terms, connects the value of the zeta series with its corresponding eta value so that,
ζ(s) = η(s) * 1/(1 – 21 – s) .
The consequent zeta result then appears fully intuitive from the conventional analytic perspective, when the eta result conforms to a similar intuition.
So for example,
η(2) = 1 – 1/4 + 1/9 – 1/16 + … = π2/12 = .8224…
And ζ(2) = η(2) * 1/(1 – 21 – s) = π2/12 * 1/(1/2) = π2/6 = 1.6449…
So both convergent results here appear intuitively correct from the standard rational perspective.
In like manner,
ζ(0) = η(0) * 1/(1 – 21 – s) = 1/2 * 1/(1 – 2) = – 1/2.
However, because η(0) does not accord with the conventional linear interpretation of the sum of a series, the result for ζ(0), therefore appears non-intuitive from this perspective.
Once again, as we have seen ζ(– 1) bears a complementary relationship with ζ(2).
However because the result for ζ(2) represents the square of a transcendental number, the resulting interpretation for ζ(– 1) - the sum of which again represents a transcendental mix of linear and circular type notions - now appears doubly non-intuitive from the standard rational perspective.
We can appreciate this better by initially looking at η(– 1) = 1 – 2 + 3 – 4 + …
And the conventional analytic sum of this series as we increase the number of terms is,
1, – 1, 2, – 2, 3, – 3, …, which diverges in conventional terms.
However, if we attempt to average the sum of these sums (as the no. of terms increases), we obtain,
1, 0, 2/3, 0, 3/5, 0, 4/7, 0, 5/9, 0,
So, as we increase the no. of terms, the resulting average will eventually alternate as between a number approximating 1/2 and 0. Note clearly that we have abandoned any strict linear notion of the value of the series as representing the quantitative sum of its independent terms!
For in now using the notion of an average to obtain the sum of a series, we are in fact mixing whole and part notions (that are circular and linear with respect to each other).
Then, once again attaining the average of these latter values (as the no. of newly acquired terms increases), the answer will converge to 1/4.
So, therefore the transformation from the eta to corresponding zeta function,
i.e. ζ(–1) = η(– 1) * 1/(1 – 21 + 1) = 1/4 * 1/(1 – 4) = – 1/12, now appears doubly non-intuitive from the standard analytic perspective.
And as the (even) value of s increases with respect to ζ(s), the non-accessibility of interpretation (in rational analytic terms) for corresponding values of ζ(1 – s) likewise increases. And this is because an increasingly holistic (where whole and part notions are continually combined) - rather than analytic - interpretation is now required.
All the values on the Riemann zeta function, for ζ(1 – s), where s > 1, appear non-intuitive (and thereby meaningless) in terms of the standard analytic interpretation of a series, i.e. where the whole is interpreted in quantitative terms as the reduced sum of its independent parts.
However, when one equally recognises both the analytic and holistic interpretation of number (in terms of number independence and number interdependence, respectively) its true nature is revealed as interactive in a complementary manner.
Therefore, ζ(s) and ζ(1 – s) are related to each other in a dynamic complementary fashion, as analytic to holistic and holistic to analytic respectively.
When an analytic type interpretation for quantitative values is appropriate for ζ(s) i.e. where s > 1, a holistic type interpretation (entailing notions of number interdependence) is then required, in relative terms, to provide intuitive meaning for interpretation of corresponding quantitative values of ζ(1 – s).
And when number is of the standard rational form for the quantitative sums of series’ values of ζ(1 – s), then in relative terms, it is irrational (transcendental) for ζ(s).
So two-way relativity of both analytic and holistic aspects is thereby involved in both quantitative and qualitative terms with respect to the interpretation of ζ(s) and ζ(1 – s).
Then in decimal terms, we have .142857 …, which
keeps recurring in 6 digit cycles.
So this is a well-known example of a cyclic
prime.
However the crucial holistic significance of this behaviour is completely missed from the conventional perspective.
So we start with the number 7 i.e. 71 as a prime, considered as a “building block” of the natural number system in 1-dimensional terms. In this sense (because 7 contains no sub-factors) it represents an example of the most independent type of number from a linear perspective.
But when we obtain the reciprocal of 7, i.e. 7 – 1, we now obtain an example of the most circular of numbers (i.e. a cyclic prime) where a high level of interdependence attaches to its unique 6-digital sequence. For example, when we multiply 142857 by 1, 2, 3, 4, 5 and 6 respectively, we derive the same circular sequence of digits.
The holistic significance of this could not be more important, as it implies that from the unconscious perspective, prime numbers are revealed as the most circular i.e. interdependent of all numbers.
And we have already explained this interdependence, in terms of the special qualitative uniqueness of its ordinal members, which indirectly can be expressed in quantitative terms as equidistant points on the circle of unit radius (in the complex plane).
Thus again as a “building block”, a prime appears as the most independent of all numbers in a quantitative (analytic) fashion; however, considered as a group comprising its ordinal natural number members, a prime by contrast appears as the most interdependent of all numbers in a qualitative (holistic) manner.
And these two complementary aspects of the primes - quantitative and qualitative - require explicit interpretation with respect to both the conscious (+) and unconscious (–) aspects of understanding, in a dynamic interactive manner.
The other key holistic explanation necessary in
this context relates to the imaginary notion
In conventional analytic terms, i relates to the square root of – 1. So when x2 = – 1,
x = i (strictly + i or – i). So what we have here is an attempt to express – 1, which relates to a 2-dimensional definition, in a reduced 1-dimensional manner.
In holistic terms – 1 (as we have seen) represents the unconscious aspect of the understanding of 1. However this causes a switch to 2-dimensional understanding, where phenomena are now understood as having two polar opposite meanings, which are dynamically interdependent with each other. Thus in a dynamic sense, negation necessarily implies an - already posited - object in an intuitive fusion of psycho-spiritual energy.
So in this context, i represents the indirect attempt to convey holistic meaning in a reduced linear rational manner. And this holistic interpretation of the imaginary notion is precisely what I have been illustrating throughout this contribution!
Thus whereas the standard analytic, represents - in qualitative terms - the real aspect of mathematical understanding, the (unrecognised) holistic aspect, indirectly conveyed in a linear rational manner, represents the corresponding imaginary aspect.
So, just as numbers can be given real and imaginary interpretations in an analytic (quantitative) manner, likewise, numbers can be given real and imaginary interpretations in a corresponding holistic (qualitative) fashion.
And a comprehensive interpretation of the Riemann zeta function (which is defined with respect to the complex plane) thereby requires both analytic and holistic interpretation in a dynamic interactive manner i.e. where numbers can have a real or imaginary meaning in both quantitative and qualitative terms.
I will just briefly attempt here to indicate the enormous significance of the holistic nature of the complex plane.
From an analytic (quantitative) perspective, the one value for which the Riemann zeta function is undefined is where s (representing an exponent, i.e. dimensional number) = 1.
It is quite similar from a holistic (qualitative) perspective, where again the one value for which the Riemann zeta function (Zeta 1 function) remains undefined is where s = 1.
And it is precisely the same for the infinite version of the Zeta 2 function (which is dynamically interdependent with the Zeta 1).
So 1 + x1 + x2 + x3 + … = 1/(1 – x)
Thus when x = 1, we obtain an infinite value, which means that the function is not defined for this value.
However the holistic interpretation is very revealing, for 1 in this context represents 1-dimensional i.e. linear interpretation, which defines the conventional mathematical approach!
Therefore, it clearly entails that from this enlarged dynamic perspective - and it is necessary to pause here to fully grasp the significance of what follows - the Riemann zeta function (and associated Zeta 2 function) cannot be properly defined in the conventional mathematical manner, where number is considered as absolute.
Thus the true nature of number is relative in a dynamic interactive fashion, with conventional mathematics representing but a special - though necessarily limited - case with interpretation taking place in 1-dimensional terms (i.e. where the qualitative aspect of relationships is reduced in a quantitative manner). Here a solely analytic (quantitative) interpretation of number is possible.
However for all other dimensional numbers (≠ 1), both analytic and holistic interpretations of number exist in a relative manner.
And from a dynamic interactive perspective, the Riemann zeta function is only properly defined for all these relative interpretations!
One remarkable example of this relative complementarity - as already mentioned - is provided through Riemann’s important functional equation, where for any value of the function ζ(s), for a positive s (> 1) on the real right hand axis, a corresponding value of the function can be given for 1 – s on the left hand axis.
So for example, if we know the value for s = 2 i.e. ζ(2), we can then, through Riemann’s equation, calculate the corresponding value for ζ(– 1).
And is well known, the value for, ζ(2) = 1 + 1/22 + 1/32 + 1/42 + … = π2/6 = 1.6449…
Now what is interesting about this value that it represents in quantitative terms, an irrational (transcendental) number with a standard rational interpretation (from a qualitative perspective).
So, the convergent value of the expression conforms to the standard rational interpretation of the value of such a series.
And in such standard (1-dimensional) interpretation, each successive term is treated (in a reduced manner) as a point on the real number line. Then the total value of the series is understood as the sum of successive part values (for each term).
Therefore, the whole is expressed in a reduced manner as the sum of its successive parts.
However though each term represents a rational number, the sum of terms is irrational (i.e. transcendental). Thus, clearly there is an unexplained qualitative dimension to the very form of a transcendental number (that reflects the combination of finite with infinite notions).
So once more, ζ(2) has an irrational (transcendental) value in quantitative terms, with a standard rational interpretation from a qualitative perspective.
Now, the key to properly appreciating the corresponding value of ζ(1 – 2), i.e. ζ(– 1) is that the two values i.e. ζ(2) and ζ(– 1) are linked in a dynamic complementary manner.
Thus, whereas the value of ζ(2) i.e. π2/6, represents an irrational (transcendental), by contrast the value of ζ(– 1) = – 1/12, represents a rational value in quantitative terms.
However ζ(– 1) = 1 + 2 + 3 + 4 + …, which from the standard rational perspective = ∞.
So clearly, we cannot interpret its result (i.e. – 1/12) with respect to the Riemann zeta function in the accepted rational manner.
Rather, because of the complementary nature of results connected through Riemann’s function, the interpretation we should now give to ζ(– 1) = – 1/12, is of an irrational (transcendental) form in qualitative terms!
We can get a deep insight into what this means through first looking at the nature of gπ from an analytic perspective, which represents the important relationship as between the circular circumference and its line diameter.
In a corresponding holistic manner, π represents an important relationship as between circular and linear type understanding (i.e. where both are in a necessary balance with each other)
Therefore one has to abandon here the linear intuition that the total quantitative value of a series represents the reduced sum of its separate (independent) terms, which again conforms to standard rational type interpretation.
Rather we must now recognise that in summing a series, both analytic notions of (linear) independence and holistic notions of (circular) interdependence are mutually involved.
To illustrate this, it is perhaps better to start with the slightly simpler value (according to the Riemann zeta function) for ζ(0) = – 1/2.
ζ(0) = 1 + 1 + 1 + …, which from the conventional analytic perspective leads to an infinite result for the series.
Therefore the obtained result i.e. – 1/2 appears non-intuitive in conventional rational terms.
However we can make progress by looking initially at the alternating eta series, i.e.
η(0) = 1 – 1 + 1 – 1 + …
In summing this series, if we take an even number of terms (where terms are naturally arranged in circular fashion as complementary couplets) the result = 0.
However if we take an odd number of terms (where each is treated as independent in a linear manner) the result = 1.
And accepting an equal probability that the series contains an even or odd number of terms respectively, we could therefore express the sum of this eta series as the average of the two possible results (1 and 0) = 1/2.
However in doing this, we thereby abandon a strict linear approach, to obtain an answer that involves in effect a balanced compromise as between linear and circular notions.
Then a simple transformation in linear terms, connects the value of the zeta series with its corresponding eta value so that,
ζ(s) = η(s) * 1/(1 – 21 – s) .
The consequent zeta result then appears fully intuitive from the conventional analytic perspective, when the eta result conforms to a similar intuition.
So for example,
η(2) = 1 – 1/4 + 1/9 – 1/16 + … = π2/12 = .8224…
And ζ(2) = η(2) * 1/(1 – 21 – s) = π2/12 * 1/(1/2) = π2/6 = 1.6449…
So both convergent results here appear intuitively correct from the standard rational perspective.
In like manner,
ζ(0) = η(0) * 1/(1 – 21 – s) = 1/2 * 1/(1 – 2) = – 1/2.
However, because η(0) does not accord with the conventional linear interpretation of the sum of a series, the result for ζ(0), therefore appears non-intuitive from this perspective.
Once again, as we have seen ζ(– 1) bears a complementary relationship with ζ(2).
However because the result for ζ(2) represents the square of a transcendental number, the resulting interpretation for ζ(– 1) - the sum of which again represents a transcendental mix of linear and circular type notions - now appears doubly non-intuitive from the standard rational perspective.
We can appreciate this better by initially looking at η(– 1) = 1 – 2 + 3 – 4 + …
And the conventional analytic sum of this series as we increase the number of terms is,
1, – 1, 2, – 2, 3, – 3, …, which diverges in conventional terms.
However, if we attempt to average the sum of these sums (as the no. of terms increases), we obtain,
1, 0, 2/3, 0, 3/5, 0, 4/7, 0, 5/9, 0,
So, as we increase the no. of terms, the resulting average will eventually alternate as between a number approximating 1/2 and 0. Note clearly that we have abandoned any strict linear notion of the value of the series as representing the quantitative sum of its independent terms!
For in now using the notion of an average to obtain the sum of a series, we are in fact mixing whole and part notions (that are circular and linear with respect to each other).
Then, once again attaining the average of these latter values (as the no. of newly acquired terms increases), the answer will converge to 1/4.
So, therefore the transformation from the eta to corresponding zeta function,
i.e. ζ(–1) = η(– 1) * 1/(1 – 21 + 1) = 1/4 * 1/(1 – 4) = – 1/12, now appears doubly non-intuitive from the standard analytic perspective.
And as the (even) value of s increases with respect to ζ(s), the non-accessibility of interpretation (in rational analytic terms) for corresponding values of ζ(1 – s) likewise increases. And this is because an increasingly holistic (where whole and part notions are continually combined) - rather than analytic - interpretation is now required.
All the values on the Riemann zeta function, for ζ(1 – s), where s > 1, appear non-intuitive (and thereby meaningless) in terms of the standard analytic interpretation of a series, i.e. where the whole is interpreted in quantitative terms as the reduced sum of its independent parts.
However, when one equally recognises both the analytic and holistic interpretation of number (in terms of number independence and number interdependence, respectively) its true nature is revealed as interactive in a complementary manner.
Therefore, ζ(s) and ζ(1 – s) are related to each other in a dynamic complementary fashion, as analytic to holistic and holistic to analytic respectively.
When an analytic type interpretation for quantitative values is appropriate for ζ(s) i.e. where s > 1, a holistic type interpretation (entailing notions of number interdependence) is then required, in relative terms, to provide intuitive meaning for interpretation of corresponding quantitative values of ζ(1 – s).
And when number is of the standard rational form for the quantitative sums of series’ values of ζ(1 – s), then in relative terms, it is irrational (transcendental) for ζ(s).
So two-way relativity of both analytic and holistic aspects is thereby involved in both quantitative and qualitative terms with respect to the interpretation of ζ(s) and ζ(1 – s).
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