If we just focus on the absolute value of the denominator of the Riemann's Zeta Function for s = - 1, - 3, - 5, - 7 and - 9, which are 12, 120, 252, 240 and 132 respectively we can find an interesting square connection.
So each of these numbers can be expressed as the product of two numbers which differ in ascending order by consecutive powers of 2.
So 12 = 4 * 3 with the difference (1) = 2^0.
120 = 12 * 10 with the difference (2) = 2^1.
252 = 18 * 14 with the difference (4) = 2^2.
240 = 20 * 12 with the difference (8) = 2^3.
132 = 22 * 6 with the difference (16) = 2^4.
After this the pattern begins to break down!
The absolute value corresponding to the denominator for s = - 11 = 32760. This can indeed be expressed as the product of two numbers that differ by a square of 2 but not (but not 2^5).
So 32760 = 182 * 180 with the difference (2) = 2^1.
However with the next number corresponding to s = - 13, no such relationship exists as between the product of two numbers i.e. involving the difference of a square of 2.
Friday, November 18, 2011
Sunday, November 6, 2011
Finite and Infinite
I have stated many times that conventional mathematical appreciation is based on a merely reduced notion of the infinite (where effectively it is treated as an extension of the finite).
This for example defines the nature of conventional proof where what is true for the general case (potentially applying to the infinite) is thereby assumed to apply to all actual (finite) cases.
And I have referred to this basic reductionism as interpretation that is linear (i.e. 1-dimensional) in qualitative terms.
This linear approach is also very much to the fore in the treatment of series where in most cases once again a seemingly unambiguous relationship as between finite and infinite emerges.
For example from a finite perspective we can see that a geometrical series such as 1 + 1/2 + 1/4 + ... converges towards some finite limit (getting ever closer to 2 without actually reaching this limiting value).
Therefore when we say that in the limit the value of the series = 2 (where the no. of terms is infinite) this again seems to comply with linear type interpretation (i.e. where the infinite is treated as an extension of finite notions).
However there are other cases where what appears true in finite terms does not readily comply with infinite notions.
For example
1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where the number of terms in linear terms is assumed infinite).
This would imply therefore that when x = - 2 that
1/3 = 1 - 2 + 4 - 8 + .....
However when we view the R.H.S in finite terms, we can see that the terms get progressively larger with the value of the series diverging. Now it is true that the terms (and consequent sum of terms) alternate between positive and negative values. However from a linear perspective we cannot say that its sum will converge to a definite finite value.
However the - apparent - equivalent L.H.S expression suggests that this is precisely what happens.
So once again from a linear perspective we obtain a result that is intuitively not in keeping with its rational mode of interpretation.
This strongly suggests that a different form of rational interpretation is required to explain the nature of the result.
However because it is qualitatively defined in terms of 1-dimensional interpretation, Conventional (Type 1) Mathematics is not appropriate for this task.
Now when we return to our example we can see what is the problem
If we consider 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1), where the series is defined in terms of a finite number (n) of terms,
then 1 - {x ^(n - 1)}/(1 - x) = 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1)
So the conclusion that
1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where n is infinite), is based on the the assumption that n - 1 = n terms (when n is infinite).
Thus the logic that applies for the infinite case i.e. n - 1 = n is directly confused with standard finite logic i.e. n - 1 ≠ n.
This then leads to the non-intuitive result (in linear rational terms) that for example
1 - 2 + 4 - 8 + .... = 1/3
Though mathematicians are of course aware of this anomaly, they attempt to explain it in terms of two results that comply with differing domains of definition.
However this avoids the deeper qualitative question of what such non-intuitive results actually entail!
Facing up to this issue requires accepting the radical conclusion that just as numbers such as 1, 2, 3, 4, ... etc. have a well-defined meaning in quantitative terms, equally they have an - as yet - unrecognised meaning in qualitative terms whereby they refer to unique modes of rational interpretation of symbols.
Once again Conventional (Type 1) Mathematics attempts to confine interpretation to 1-dimensional logic in qualitative terms. However potentially an unlimited set of other qualitative logical interpretations can be given.
And the series that I have used to illustrate this point itself points to the need for a different means of rational interpretation (so that results can then intuitively concur with the correct rational mode adopted).
This for example defines the nature of conventional proof where what is true for the general case (potentially applying to the infinite) is thereby assumed to apply to all actual (finite) cases.
And I have referred to this basic reductionism as interpretation that is linear (i.e. 1-dimensional) in qualitative terms.
This linear approach is also very much to the fore in the treatment of series where in most cases once again a seemingly unambiguous relationship as between finite and infinite emerges.
For example from a finite perspective we can see that a geometrical series such as 1 + 1/2 + 1/4 + ... converges towards some finite limit (getting ever closer to 2 without actually reaching this limiting value).
Therefore when we say that in the limit the value of the series = 2 (where the no. of terms is infinite) this again seems to comply with linear type interpretation (i.e. where the infinite is treated as an extension of finite notions).
However there are other cases where what appears true in finite terms does not readily comply with infinite notions.
For example
1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where the number of terms in linear terms is assumed infinite).
This would imply therefore that when x = - 2 that
1/3 = 1 - 2 + 4 - 8 + .....
However when we view the R.H.S in finite terms, we can see that the terms get progressively larger with the value of the series diverging. Now it is true that the terms (and consequent sum of terms) alternate between positive and negative values. However from a linear perspective we cannot say that its sum will converge to a definite finite value.
However the - apparent - equivalent L.H.S expression suggests that this is precisely what happens.
So once again from a linear perspective we obtain a result that is intuitively not in keeping with its rational mode of interpretation.
This strongly suggests that a different form of rational interpretation is required to explain the nature of the result.
However because it is qualitatively defined in terms of 1-dimensional interpretation, Conventional (Type 1) Mathematics is not appropriate for this task.
Now when we return to our example we can see what is the problem
If we consider 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1), where the series is defined in terms of a finite number (n) of terms,
then 1 - {x ^(n - 1)}/(1 - x) = 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1)
So the conclusion that
1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where n is infinite), is based on the the assumption that n - 1 = n terms (when n is infinite).
Thus the logic that applies for the infinite case i.e. n - 1 = n is directly confused with standard finite logic i.e. n - 1 ≠ n.
This then leads to the non-intuitive result (in linear rational terms) that for example
1 - 2 + 4 - 8 + .... = 1/3
Though mathematicians are of course aware of this anomaly, they attempt to explain it in terms of two results that comply with differing domains of definition.
However this avoids the deeper qualitative question of what such non-intuitive results actually entail!
Facing up to this issue requires accepting the radical conclusion that just as numbers such as 1, 2, 3, 4, ... etc. have a well-defined meaning in quantitative terms, equally they have an - as yet - unrecognised meaning in qualitative terms whereby they refer to unique modes of rational interpretation of symbols.
Once again Conventional (Type 1) Mathematics attempts to confine interpretation to 1-dimensional logic in qualitative terms. However potentially an unlimited set of other qualitative logical interpretations can be given.
And the series that I have used to illustrate this point itself points to the need for a different means of rational interpretation (so that results can then intuitively concur with the correct rational mode adopted).
Wednesday, November 2, 2011
The Strange Case of η( - 1)
The Eta Series for s = - 1 is,
η( - 1) = 1/1^(- 1) - 1/2^(- 1) + 1/3^(- 1) - 1/4^(- 1) + ....
Therefore
η( - 1) = 1 - 2 + 3 - 4 + 5 - 6 + ....
Withe reference to the Riemann Zeta Function the value for this Eta series = 1/4.
The question then arises as what meaning can we give this result!
It is perhaps better in illustrating to start with η(0) = 1 - 1 + 1 - 1 + 1 - 1 + ...
The value of this alternating series = 1/2
It is easy enough in this case to see how this value might arise!
If we take an even number of terms, the sum of the series = 0.
However if we take an odd number of terms the sum = 1.
Therefore it seems reasonable - where the number of terms is unspecified - to average the two values as 1/2. However in illustrating this we need to consider a finite number of terms.
In more general terms the answer here is n/2 (where the nth term when it is odd = 1).
However what is interesting is that when we now consider the series (in Type 1 mathematical terms) as infinite
i.e. 1/(1 + x) = 1 + x + x^2 + x^3 + x^4 + ...,
by setting x = - 1 we obtain 1/2 unambiguously as the correct answer.
i.e. 1/2 = 1 - 1 + 1 - 1 + 1 - ..
So the value here from the infinite perspective as 1/2 can be simply considered as n/2 (where the quantitative value of n is set at a default value of 1).
What I am getting at here is very significant indeed in terms of properly interpreting the nature of the Riemann Hypothesis!
Conventional (Type 1) Mathematics is inherently defined with respect to a default dimensional (qualitative) value of 1. In other words the nature of Type 1 Mathematics is qualitatively linear (1-dimensional) where all quantitative values are ultimately reduced in 1-dimensional terms.
So for example 2^2 = 4^1 (in Type 1 terms).
However Holistic (Type 2) Mathematics - in inverse fashion - is inherently defined with respect to a default base quantitative value of 1. So in concentrating on the nature of qualitative transformation (as with the switch from finite to infinite series) Type 2 interpretation is not directly concerned with the quantitative nature of number but rather as its representation of a holistic transformation (where dimensional powers other than + 1 are entailed)!
Now this will perhaps become clearer when we look at both the finite and infinite interpretation of terms corresponding to η(- 1).
η(- 1) = 1 - 2 + 3 - 4 + .....
We consider this series initially as finite and attempt to sum its value in linear (Type 1) terms.
For example we will initially derive η(0) with 10 terms.
i.e. y = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9
Then when we differentiate y with respect to x we obtain 9 remaining terms on the RHS
i.e. 1 - 2x + 3(x^2) - 4(x^3) + 5(x^4) - 6(x^5) + 7(x^6) - 8(x^7) + 9(x^8)
Setting x = - 1 we obtain
1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9
Then in summing this finite series by grouping terms in pairs
the sum of the first 8 terms is - 1 - 1 - 1 - 1
So, if the number of terms is even the sum of the series = - (n - 2)/2
Thus as we originally started with n = 10 the sum of the first 8 terms = - (8/2) = - 4.
However if we sum the first n - 1 terms (which is now odd) we obtain - (n - 2)/2 + (n - 1).
Thus the sum of the first 9 terms = - 4 + 9 = 5.
As we did before for η(0), since there is a 50:50 chance of obtaining the positive value for the sum (associated with an odd number of terms). So the average = {(n - 1) - (n - 2)/2}/2 = (2n - 2 - n + 2)/4 = n/4
So again with originally n = 10, the sum of the first 9 terms = 5. As there is a 50:50 chance of getting this value, the expected value is therefore 2.5. And this is the value corresponding to n/4 (where n = 10).
What is fascinating is that when we then consider the series in infinite terms through differentiating both sides of original expression we get
1/(1 + x)^2 = 1 + 2^x + 3(x^2) + 4(x^3) + .....
By setting x = - 1,
1/4 = 1 - 2 + 3 - 4 + 5 - .....
So once again the sum for the infinite series is the same as derived for the finite, with the important difference that n is here given a default value of 1.
This strongly suggests that in transforming from finite to infinite expressions the very representation of number itself switches from a specific (quantitative) to a holistic (qualitative) meaning. In other words the expected value of the nth term (when n is odd) is n/4. So when referring to the general ratio (rather than its specific quantitative value) we get 1/4!
η( - 1) = 1/1^(- 1) - 1/2^(- 1) + 1/3^(- 1) - 1/4^(- 1) + ....
Therefore
η( - 1) = 1 - 2 + 3 - 4 + 5 - 6 + ....
Withe reference to the Riemann Zeta Function the value for this Eta series = 1/4.
The question then arises as what meaning can we give this result!
It is perhaps better in illustrating to start with η(0) = 1 - 1 + 1 - 1 + 1 - 1 + ...
The value of this alternating series = 1/2
It is easy enough in this case to see how this value might arise!
If we take an even number of terms, the sum of the series = 0.
However if we take an odd number of terms the sum = 1.
Therefore it seems reasonable - where the number of terms is unspecified - to average the two values as 1/2. However in illustrating this we need to consider a finite number of terms.
In more general terms the answer here is n/2 (where the nth term when it is odd = 1).
However what is interesting is that when we now consider the series (in Type 1 mathematical terms) as infinite
i.e. 1/(1 + x) = 1 + x + x^2 + x^3 + x^4 + ...,
by setting x = - 1 we obtain 1/2 unambiguously as the correct answer.
i.e. 1/2 = 1 - 1 + 1 - 1 + 1 - ..
So the value here from the infinite perspective as 1/2 can be simply considered as n/2 (where the quantitative value of n is set at a default value of 1).
What I am getting at here is very significant indeed in terms of properly interpreting the nature of the Riemann Hypothesis!
Conventional (Type 1) Mathematics is inherently defined with respect to a default dimensional (qualitative) value of 1. In other words the nature of Type 1 Mathematics is qualitatively linear (1-dimensional) where all quantitative values are ultimately reduced in 1-dimensional terms.
So for example 2^2 = 4^1 (in Type 1 terms).
However Holistic (Type 2) Mathematics - in inverse fashion - is inherently defined with respect to a default base quantitative value of 1. So in concentrating on the nature of qualitative transformation (as with the switch from finite to infinite series) Type 2 interpretation is not directly concerned with the quantitative nature of number but rather as its representation of a holistic transformation (where dimensional powers other than + 1 are entailed)!
Now this will perhaps become clearer when we look at both the finite and infinite interpretation of terms corresponding to η(- 1).
η(- 1) = 1 - 2 + 3 - 4 + .....
We consider this series initially as finite and attempt to sum its value in linear (Type 1) terms.
For example we will initially derive η(0) with 10 terms.
i.e. y = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9
Then when we differentiate y with respect to x we obtain 9 remaining terms on the RHS
i.e. 1 - 2x + 3(x^2) - 4(x^3) + 5(x^4) - 6(x^5) + 7(x^6) - 8(x^7) + 9(x^8)
Setting x = - 1 we obtain
1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9
Then in summing this finite series by grouping terms in pairs
the sum of the first 8 terms is - 1 - 1 - 1 - 1
So, if the number of terms is even the sum of the series = - (n - 2)/2
Thus as we originally started with n = 10 the sum of the first 8 terms = - (8/2) = - 4.
However if we sum the first n - 1 terms (which is now odd) we obtain - (n - 2)/2 + (n - 1).
Thus the sum of the first 9 terms = - 4 + 9 = 5.
As we did before for η(0), since there is a 50:50 chance of obtaining the positive value for the sum (associated with an odd number of terms). So the average = {(n - 1) - (n - 2)/2}/2 = (2n - 2 - n + 2)/4 = n/4
So again with originally n = 10, the sum of the first 9 terms = 5. As there is a 50:50 chance of getting this value, the expected value is therefore 2.5. And this is the value corresponding to n/4 (where n = 10).
What is fascinating is that when we then consider the series in infinite terms through differentiating both sides of original expression we get
1/(1 + x)^2 = 1 + 2^x + 3(x^2) + 4(x^3) + .....
By setting x = - 1,
1/4 = 1 - 2 + 3 - 4 + 5 - .....
So once again the sum for the infinite series is the same as derived for the finite, with the important difference that n is here given a default value of 1.
This strongly suggests that in transforming from finite to infinite expressions the very representation of number itself switches from a specific (quantitative) to a holistic (qualitative) meaning. In other words the expected value of the nth term (when n is odd) is n/4. So when referring to the general ratio (rather than its specific quantitative value) we get 1/4!
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