As Euler showed the harmonic series 1 + 1/2 + 1/3 + 1/4 +..... + n is approximated as log n + λ (for a finite value of n).
And as λ (the Euler-Masceroni constant) is a constant = .5772 approx this means that for large n, log n is approximated by the harmonic series.
This means therefore that perhaps the simplest expression for the frequency of prime distribution is given as the sum of the harmonic series for n terms divided by n (which becomes increasingly accurate for larger n).
Looked at another way the sum of the harmonic series (for large n) approximates the average spread or gap as between prime numbers in the region of n.
Therefore as the sum of the first million terms for example of the harmonic series = 14.384 (approx). Therefore the average gap as between primes in the region of 1,000,000 is roughly 14. Though this approximation is not yet very accurate, the approximation would greatly improve (in relative terms) as the value of n increases.
Now it is well known that the average spread as between primes continually increases as the value of n increases.
What I find particularly striking in this regard is that the increase in the average spread (or gap) as between primes as we increase n by 1 is given by 1/n.
So for example as we increase n 1,000,000 to 1,000,001 the increase in the average gap as between primes is 1/1,000,000.
(More accurately as we increase n from 999,999.5 to 1,000,000.5 the average gap between primes increases by 1/1,000,000).
This result can easily be demonstrated through differentiation of log n + λ (with respect to n) which results in 1/n.
Now if we multiply the simple expression for the general frequency of primes i.e. n/log n by 1/n we obtain 1/log n (which represents the probability that n is prime).
Thus, we can say that the product of the general frequency of prime distribution and the change in the average gap as between primes (for large n) approximates well the probability that n is prime.
Wednesday, December 14, 2011
Friday, November 18, 2011
Interesting Square Connection
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.
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.
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!
Monday, October 31, 2011
Calculating ζ (- 1) and ζ (- 3)
Without attempting to use analytic continuation on the complex plane, we will now calculate the first two values for negative odd integer values of s i.e. - 1 and - 3, for the Riemann Zeta Function.
1/(1 - s) = 1 + s + s^2 + s^3 + s^4 + s^5 + s^6 + .......
Differentiating both sides,
1/{(1 - s)^2} = 1 + 2s + 3(s^2) + 4(s^3) + 5(s^4) + 6(s^5) + .....
Setting s = - 1,
1/4 = 1 - 2 + 3 - 4 + 5 - 6 + ......
Therefore 1/4 = η(- 1)
ζ(- 1) = η(- 1)/{1 - 2^(s - 1)} = (1/4)/(- 3) = - 1/12
If we differentiate both sides again
2/{1 - s)^3 = 2 + 6s + 12(s^2) + 20(s^3) + 30(s^4) +....
Then differentiating once more,
6/(1 - s)^4 = 6 + 24s + 60(s^2) + 120(s^3) +.......
Setting s = - 1,
6/16 = 6 - 24 + 60 - 120 + .....
η(- 3) = 1 - 2^3 + 3^3 - 4^3 + 5^4 - ......
= 1 - 8 + 27 - 64 + 125 - ....
So η(- 3) + 6/16
= (1 - 8 + 27 - 64 + 125 - ....) + (6 - 24 + 60 - 120 + .....)
= 1 - 2 + 3 - 4 + 5 - ......
Thus η(- 3) + 6/16 = η(- 1)
Therefore η(- 3) = η(- 1) - 6/16
= 1/4 - 6/16 = 4/16 - 6/16 = - 2/16 = - 1/8
ζ(- 3) = η(- 3)/{1 - 2^(s - 3)} = (- 1/8)/(- 15) = 1/120
This would suggest that in principle we should be able to work out all the odd negative integer values for s through a similar process of combining already attained values for s with the varied series that arise from continued differentiation of both sides of the equation.
The question then arises as to why similar attempts with respect to even integer values of s do not hold.
As we have seen,
2/{1 - s)^3 = 2 + 6s + 12(s^2) + 20(s^3) + 30(s^4) - 42(s^5) + ....
Thus setting s = - 1,
2/8 = 2 - 6 + 12 - 20 + 30 - 42 + ....
Therefore by combining terms in successive pairs 2/8 = - 4 - 8 - 12 - ......
= - 4(1 + 2 + 3 + ....)
As the term inside the bracket is ζ(- 1), this would imply that
2/8 = - 4(- 1/12) = 1/3 which is meaningless.
However what is interesting in this case is that instead of obtaining an eta expression on the RHS (as with odd integer values for s) we obtain a zeta value.
So clearly whereas we can derive zeta values from eta, we cannot here derive a zeta value from another zeta value!
The reason is that this value for ζ(- 1) arises when we set s = + 1 on the RHS which means that the value for 2/{1 - s)^3 is thereby infinite!
Once again
when s = - 1,
2/8 = 2 - 6 + 12 - 20 + 30 - 42 + ....
η(- 1) = 1 - 2 + 3 - 4 + 5 - 6 + ....
Therefore 2/8 - η(- 1) = 1 - 4 + 9 - 16 + 25 - 36 + ...
So 2/8 - η(- 1) = η(- 2)
Therefore η(- 2) = 2/8 - 1/4 = 0.
And as ζ(- 2) = η(- 2)/{1 - 2^(s - 1)} this implies that ζ(- 2)= 0.
Thus we derive the correct answer in this case.
And a similar result in principle would emerge whenever we use an negative even integer value (i.e. - 4, - 6, -8, etc.) for s.
1/(1 - s) = 1 + s + s^2 + s^3 + s^4 + s^5 + s^6 + .......
Differentiating both sides,
1/{(1 - s)^2} = 1 + 2s + 3(s^2) + 4(s^3) + 5(s^4) + 6(s^5) + .....
Setting s = - 1,
1/4 = 1 - 2 + 3 - 4 + 5 - 6 + ......
Therefore 1/4 = η(- 1)
ζ(- 1) = η(- 1)/{1 - 2^(s - 1)} = (1/4)/(- 3) = - 1/12
If we differentiate both sides again
2/{1 - s)^3 = 2 + 6s + 12(s^2) + 20(s^3) + 30(s^4) +....
Then differentiating once more,
6/(1 - s)^4 = 6 + 24s + 60(s^2) + 120(s^3) +.......
Setting s = - 1,
6/16 = 6 - 24 + 60 - 120 + .....
η(- 3) = 1 - 2^3 + 3^3 - 4^3 + 5^4 - ......
= 1 - 8 + 27 - 64 + 125 - ....
So η(- 3) + 6/16
= (1 - 8 + 27 - 64 + 125 - ....) + (6 - 24 + 60 - 120 + .....)
= 1 - 2 + 3 - 4 + 5 - ......
Thus η(- 3) + 6/16 = η(- 1)
Therefore η(- 3) = η(- 1) - 6/16
= 1/4 - 6/16 = 4/16 - 6/16 = - 2/16 = - 1/8
ζ(- 3) = η(- 3)/{1 - 2^(s - 3)} = (- 1/8)/(- 15) = 1/120
This would suggest that in principle we should be able to work out all the odd negative integer values for s through a similar process of combining already attained values for s with the varied series that arise from continued differentiation of both sides of the equation.
The question then arises as to why similar attempts with respect to even integer values of s do not hold.
As we have seen,
2/{1 - s)^3 = 2 + 6s + 12(s^2) + 20(s^3) + 30(s^4) - 42(s^5) + ....
Thus setting s = - 1,
2/8 = 2 - 6 + 12 - 20 + 30 - 42 + ....
Therefore by combining terms in successive pairs 2/8 = - 4 - 8 - 12 - ......
= - 4(1 + 2 + 3 + ....)
As the term inside the bracket is ζ(- 1), this would imply that
2/8 = - 4(- 1/12) = 1/3 which is meaningless.
However what is interesting in this case is that instead of obtaining an eta expression on the RHS (as with odd integer values for s) we obtain a zeta value.
So clearly whereas we can derive zeta values from eta, we cannot here derive a zeta value from another zeta value!
The reason is that this value for ζ(- 1) arises when we set s = + 1 on the RHS which means that the value for 2/{1 - s)^3 is thereby infinite!
Once again
when s = - 1,
2/8 = 2 - 6 + 12 - 20 + 30 - 42 + ....
η(- 1) = 1 - 2 + 3 - 4 + 5 - 6 + ....
Therefore 2/8 - η(- 1) = 1 - 4 + 9 - 16 + 25 - 36 + ...
So 2/8 - η(- 1) = η(- 2)
Therefore η(- 2) = 2/8 - 1/4 = 0.
And as ζ(- 2) = η(- 2)/{1 - 2^(s - 1)} this implies that ζ(- 2)= 0.
Thus we derive the correct answer in this case.
And a similar result in principle would emerge whenever we use an negative even integer value (i.e. - 4, - 6, -8, etc.) for s.
Wednesday, September 28, 2011
New Perspective on Mathematical Proof
I have argued before that correctly speaking - in dynamic experiential terms - that all mathematical proof is subject to the uncertainty principle.
Thus it represents but an especially important form of social consensus that can never be absolute. In fact in certain terms such consensus can prove especially flawed.
Thus for a period in 1993, it was believed that that Fermat's Last theorem had been proved (only for a fatal flaw in reasoning later being discovered). Now the prevailing consensus since 1995 is that this remaining problem has been satisfactorily resolved. So as time goes by - with no further flaws being discovered - we can accept with an ever greater degree of confidence that Fermat's Last Theorem has been proven. However this conviction always remains of a merely probabilistic nature (that is necessarily subject to a degree of uncertainty).
At a deeper level I have challenged the conventional notion of mathematical proof in that it represents just one limited form of interpretation (where the qualitative aspect of understanding is necessarily reduced in quantitative terms).
More formally, in holistic mathematical terms conventional proof corresponds to a linear (1-dimensional) mode of qualitative interpretation (where again meaning is reduced in quantitative terms). However correctly understood we can have potentially an infinite set of mathematical interpretations (corresponding to all other dimensional numbers) where quantitative and qualitative aspects of interpretation - though necessarily related - preserve a distinctive aspect. And in the inevitable relative interaction between both aspects a degree of uncertainty necessarily exists.
So therefore a comprehensive proof requires both qualitative and quantitative aspects (that are necessarily relative). Conventional proof only appears therefore of an absolute nature because the qualitative aspect is entirely neglected.
However recently I have come to realise that there is yet another way in which the uncertainty principle necessarily applies to all mathematical proof.
As we have seen the Riemann Hypothesis represents the important starting axiom whereby the quantitative and qualitative aspects of mathematical understanding can be directly reconciled.
So the Riemann Hypothesis is already necessarily inherent in conventional mathematical aspects. Because of the reduced nature of understanding (where the qualitative aspect of truth is reduced to the quantitative), it is already assumed that if a theorem is proved in general (holistic) terms that it thereby necessarily applies to each individual case (within its class). So for example one we accept that the Pythagorean Theorem is true for the general case (establishing potentially that in any right angled triangle the square on the hypotenuse is equal to the sum of squares on the other two sides) that this necessarily applies in any actual case.
However this implies confusing the potentially infinite nature of the holistic general proposition with the actual finite nature of individual examples. In other words the qualitative aspect of understanding is reduced thereby to the quantitative.
Thus when we properly preserve the unique qualitative distinction of (potential) infinite and (actual) finite notions, we cannot in the absence of a higher authority as it were, automatically infer the truth of the specific from the corresponding truth of the general case.
Now acceptance of the Riemann Hypothesis is necessary to correctly make such an inference. However, as we have seen, because the Riemann Hypothesis supersedes conventional mathematical axioms it cannot be proved (or disproved) from a conventional mathematical perspective.
Therefore the acceptance that a general (qualitative) proof applies in specific (quantitative) terms is correctly based on the validity of the Riemann Hypothesis (which itself cannot be proved or disproved).
That means therefore that mathematical proof is ultimately based on an act of faith (and is thereby subject to uncertainty).
In the highly unlikely case that a non-trivial zero of the Riemann Zeta Function is ever found off the line (with real part = .5) this will pose an interesting dilemma.
We may be tempted to initially maintain that the Riemann Hypothesis has in fact been proven to be untrue.
However this would raise a much deeper problem. For if the Riemann Hypothesis is untrue, then we are no longer entitled to maintain the connection between an actual specific case and the general truth (for the potentially infinite case).
In other words we would no longer be entitled to infer that the demonstrated finding of our errant zero undermines the truth of the general proposition.
In fact it would be even much worse in that we would no longer be able to trust any proposition that has been proved in conventional terms.
So once again the mathematical edifice is ultimately dependent on a supreme act of faith (in the correspondence of infinite with finite notions).
For this reason anyone who believes in Mathematics should hope - and indeed pray - that no errant zero ever crops up for this would rightly undermine faith in the whole enterprise!
Thus it represents but an especially important form of social consensus that can never be absolute. In fact in certain terms such consensus can prove especially flawed.
Thus for a period in 1993, it was believed that that Fermat's Last theorem had been proved (only for a fatal flaw in reasoning later being discovered). Now the prevailing consensus since 1995 is that this remaining problem has been satisfactorily resolved. So as time goes by - with no further flaws being discovered - we can accept with an ever greater degree of confidence that Fermat's Last Theorem has been proven. However this conviction always remains of a merely probabilistic nature (that is necessarily subject to a degree of uncertainty).
At a deeper level I have challenged the conventional notion of mathematical proof in that it represents just one limited form of interpretation (where the qualitative aspect of understanding is necessarily reduced in quantitative terms).
More formally, in holistic mathematical terms conventional proof corresponds to a linear (1-dimensional) mode of qualitative interpretation (where again meaning is reduced in quantitative terms). However correctly understood we can have potentially an infinite set of mathematical interpretations (corresponding to all other dimensional numbers) where quantitative and qualitative aspects of interpretation - though necessarily related - preserve a distinctive aspect. And in the inevitable relative interaction between both aspects a degree of uncertainty necessarily exists.
So therefore a comprehensive proof requires both qualitative and quantitative aspects (that are necessarily relative). Conventional proof only appears therefore of an absolute nature because the qualitative aspect is entirely neglected.
However recently I have come to realise that there is yet another way in which the uncertainty principle necessarily applies to all mathematical proof.
As we have seen the Riemann Hypothesis represents the important starting axiom whereby the quantitative and qualitative aspects of mathematical understanding can be directly reconciled.
So the Riemann Hypothesis is already necessarily inherent in conventional mathematical aspects. Because of the reduced nature of understanding (where the qualitative aspect of truth is reduced to the quantitative), it is already assumed that if a theorem is proved in general (holistic) terms that it thereby necessarily applies to each individual case (within its class). So for example one we accept that the Pythagorean Theorem is true for the general case (establishing potentially that in any right angled triangle the square on the hypotenuse is equal to the sum of squares on the other two sides) that this necessarily applies in any actual case.
However this implies confusing the potentially infinite nature of the holistic general proposition with the actual finite nature of individual examples. In other words the qualitative aspect of understanding is reduced thereby to the quantitative.
Thus when we properly preserve the unique qualitative distinction of (potential) infinite and (actual) finite notions, we cannot in the absence of a higher authority as it were, automatically infer the truth of the specific from the corresponding truth of the general case.
Now acceptance of the Riemann Hypothesis is necessary to correctly make such an inference. However, as we have seen, because the Riemann Hypothesis supersedes conventional mathematical axioms it cannot be proved (or disproved) from a conventional mathematical perspective.
Therefore the acceptance that a general (qualitative) proof applies in specific (quantitative) terms is correctly based on the validity of the Riemann Hypothesis (which itself cannot be proved or disproved).
That means therefore that mathematical proof is ultimately based on an act of faith (and is thereby subject to uncertainty).
In the highly unlikely case that a non-trivial zero of the Riemann Zeta Function is ever found off the line (with real part = .5) this will pose an interesting dilemma.
We may be tempted to initially maintain that the Riemann Hypothesis has in fact been proven to be untrue.
However this would raise a much deeper problem. For if the Riemann Hypothesis is untrue, then we are no longer entitled to maintain the connection between an actual specific case and the general truth (for the potentially infinite case).
In other words we would no longer be entitled to infer that the demonstrated finding of our errant zero undermines the truth of the general proposition.
In fact it would be even much worse in that we would no longer be able to trust any proposition that has been proved in conventional terms.
So once again the mathematical edifice is ultimately dependent on a supreme act of faith (in the correspondence of infinite with finite notions).
For this reason anyone who believes in Mathematics should hope - and indeed pray - that no errant zero ever crops up for this would rightly undermine faith in the whole enterprise!
Wednesday, September 21, 2011
The Mystery of the Primes
I have been reading Matthew Watkins' book "The Mystery of the Primes" (the first volume of a proposed trilogy) recently with much interest.
Though Matthew is clearly a qualified mathematician and therefore well able to deal with issues relating to the primes in the accepted specialised language that fellow practitioners employ, he opts here for a user friendly approach that would be accessible to most lay people (with little or no grounding in Mathematics).
What I like about this approach is that he clearly appreciates how apparently simple mathematical concepts lead to profound problems of a philosophical nature. So philosophical, psychological, religious and even economic observations are introduced early on in a much more wide ranging approach than is conventional, to unveiling the mystery of the primes.
I look forward to following Matthew's trail to see where it will lead in the further two volumes (yet to be published). I am sure that it will be interesting!
Having introduced - what he refers to as - "spiral waves" he then intends to probe more deeply into their relationship to the famed Riemann zeros in the second volume. These, magically have the capacity to eliminate the deviations associated with the generalised prediction of prime number frequency (relating to a continuous function) so that ultimately we can precisely obtain the actual frequency of such primes (which is of a discrete nature).
Such deviations seemingly correspond precisely to the vibrations of some unknown system (with a tantalising relationship to known quantum chaotic systems of a physical nature).
So a fundamental question relates to what in fact is vibrating!
My own considerations on this issue have led to the conclusion that it is the prime numbers themselves that are vibrating.
We are still far too accustomed to think of prime numbers in discrete independent terms as the building blocks of the number system. However, even momentary reflection on the matter reveals that they can have no strict meaning in the absence of the natural numbers.
To resolve this issue one must therefore look at both the prime and the natural numbers in a dynamic interactive manner. Implicit in this observation is the clear recognition that numbers have both quantitative and qualitative aspects. So rather than having a static identity (of an abstract nature) all numbers - from this new perspective - represent dynamic interaction patterns (with both quantitative and qualitative characteristics).
To be precise, the prime number code - which governs subsequent phenomenal interactions - is of a potential kind and thereby already inherent in the physical phenomena that emerge in reality. However the very unfolding of such phenomena in space and time implies some degree of separation of the quantitative from the qualitative aspect of the primes. Once again therefore the ultimate state where quantitative and qualitative (or specific and holistic) aspects are perfectly reconciled (in direct spiritual union of reality) is of an absolute ineffable nature.
And it is to this ultimate state that the Riemann Hypothesis directly points! In other words the Riemann Hypothesis relates to the fundamental condition for maintaining consistency with respect to both the quantitative and qualitative behaviour of the primes. Without such consistency (which unfortunately is the case if the Riemann Hypothesis does not hold) we would have no basis for either trusting the validity of our mathematical axioms or our holistic intuitions as to nature of mathematical truth.
As the condition, to which the Riemann Hypothesis relates, precedes such axioms and intuitions (and is already inherent in their very use) there is no way in which the Riemann Hypothesis can be proved (or disproved) in conventional mathematical terms.
It is therefore truly of the most fundamental nature possible, with its acceptance representing an act of faith in the subsequent consistency of the whole mathematical enterprise!
So in a qualified sense the prime numbers themselves vibrate! More correctly it is the interaction of both the prime and natural numbers that leads to the vibrations corresponding to the Riemann non-trivial zeros. Once again though the starting code is set in an absolute ineffable manner, these vibrations can only become manifest in a phenomenal physical manner (where some degree of separation of both quantitative and qualitative aspects has already taken place).
So the fascinating conclusion that we can draw from this is that manifest physical reality itself represents, at a more fundamental level, vibrations corresponding to number patterns (which are thereby inherent in all physical forms).
Now this might appear ludicrous when we think of numbers with respect to their mere quantitative characteristics. However when we equally recognise the qualitative dimensional characteristics (associated with number) it seems - at least to me - a somewhat obvious conclusion.
And this is the precise nub of the matter. Properly understood the dimensional aspect (i.e. power or exponent) of a number is of a qualitative nature (with respect to its number quantity). However this simple truth is - seemingly - entirely missed in conventional Type 1 appreciation. So when the number a is raised to a corresponding dimensional number b (as power or exponent) i.e. a^b, then - properly understood a and b are quantitative and qualitative with respect to each other.
A simple clue to this truth is given by the fact that when 1 is raised to a dimension that is a rational fraction such as 1/3, a circular number results (i.e. lying on the circle of unit radius in the complex plane). So though 1 and 1/3 are conventionally interpreted as linear quantities (i.e. as points on the real number line) clearly a profound transformation results (from linear to circular form) in relating the base quantity 1 with its dimensional exponent 1/3. So the simple truth is that these numbers - which can indeed be given a quantitative definition in isolation - are actually quantitative and qualitative in relation to each other!
And Type 2 i.e. Holistic Mathematics (which is directly of a qualitative nature) is based on circular rather than linear logical notions!
One important implication of this finding - which I was even dimly aware of as a child - is that one cannot hope to give a coherent explanation of an apparently simple relationship such as the square root of 1 in the absence of Type 2 (qualitative) mathematical understanding!
So remarkably, because of the unquestioned quantitative bias of Conventional Mathematics (which is thereby quite unbalanced) the most obvious problems of mathematical interpretation are persistently overlooked!
Therefore, for example when a number is raised to a whole number dimension (other than 1) in conventional terms a merely reduced quantitative interpretation is given of the result.
Thus 2^3 in Type 1 terms = 8 (i.e. 8^1).
Now by looking at this in a geometrical manner we can easily appreciate that a dimensional transformation of a qualitative nature is equally involved. In other words 2^3 represents cube (3-dimensional) rather than linear (1-dimensional) units.
So, as I have repeatedly expressed, Conventional (Type 1) Mathematics necessarily reduces the qualitative notion of dimension in a merely linear (1-dimensional) fashion.
However, when we begin to appreciate the true qualitative nature of dimension, we perhaps can recognise that physical dimensions - when correctly understood - are intimately related to holistic mathematical notions (of a qualitative Type 2 nature).
I have expressed for some time - see my Integral Science blog - dissatisfaction with the philosophical notion of a string (which has no strict physical meaning).
However once we accept that physical phenomena represent a certain rigid reduction with respect to original vibration patterns of a purely mathematical nature, then we can perhaps appreciate the ultimate nature of strings as the prime number constituents of natural reality with the relationship collectively between strings equally representing the prime number dimensional aspects of this same natural reality.
Reality at its most fundamental level is written in a double binary code (1 and 0) where both quantitative and qualitative aspects interact. This provides a means therefore of potentially encoding all (quantitative) information as is well recognised in the present digital age; however what is not equally recognised is that the same digits when holistically interpreted in a qualitative manner can potentially encode all transformation processes!
Then at the next most fundamental level, reality is written in the prime number code (again with respect to its quantitative and qualitative aspects). And from this code ultimately all phenomenal reality, as we know it, is derived!
Though Matthew is clearly a qualified mathematician and therefore well able to deal with issues relating to the primes in the accepted specialised language that fellow practitioners employ, he opts here for a user friendly approach that would be accessible to most lay people (with little or no grounding in Mathematics).
What I like about this approach is that he clearly appreciates how apparently simple mathematical concepts lead to profound problems of a philosophical nature. So philosophical, psychological, religious and even economic observations are introduced early on in a much more wide ranging approach than is conventional, to unveiling the mystery of the primes.
I look forward to following Matthew's trail to see where it will lead in the further two volumes (yet to be published). I am sure that it will be interesting!
Having introduced - what he refers to as - "spiral waves" he then intends to probe more deeply into their relationship to the famed Riemann zeros in the second volume. These, magically have the capacity to eliminate the deviations associated with the generalised prediction of prime number frequency (relating to a continuous function) so that ultimately we can precisely obtain the actual frequency of such primes (which is of a discrete nature).
Such deviations seemingly correspond precisely to the vibrations of some unknown system (with a tantalising relationship to known quantum chaotic systems of a physical nature).
So a fundamental question relates to what in fact is vibrating!
My own considerations on this issue have led to the conclusion that it is the prime numbers themselves that are vibrating.
We are still far too accustomed to think of prime numbers in discrete independent terms as the building blocks of the number system. However, even momentary reflection on the matter reveals that they can have no strict meaning in the absence of the natural numbers.
To resolve this issue one must therefore look at both the prime and the natural numbers in a dynamic interactive manner. Implicit in this observation is the clear recognition that numbers have both quantitative and qualitative aspects. So rather than having a static identity (of an abstract nature) all numbers - from this new perspective - represent dynamic interaction patterns (with both quantitative and qualitative characteristics).
To be precise, the prime number code - which governs subsequent phenomenal interactions - is of a potential kind and thereby already inherent in the physical phenomena that emerge in reality. However the very unfolding of such phenomena in space and time implies some degree of separation of the quantitative from the qualitative aspect of the primes. Once again therefore the ultimate state where quantitative and qualitative (or specific and holistic) aspects are perfectly reconciled (in direct spiritual union of reality) is of an absolute ineffable nature.
And it is to this ultimate state that the Riemann Hypothesis directly points! In other words the Riemann Hypothesis relates to the fundamental condition for maintaining consistency with respect to both the quantitative and qualitative behaviour of the primes. Without such consistency (which unfortunately is the case if the Riemann Hypothesis does not hold) we would have no basis for either trusting the validity of our mathematical axioms or our holistic intuitions as to nature of mathematical truth.
As the condition, to which the Riemann Hypothesis relates, precedes such axioms and intuitions (and is already inherent in their very use) there is no way in which the Riemann Hypothesis can be proved (or disproved) in conventional mathematical terms.
It is therefore truly of the most fundamental nature possible, with its acceptance representing an act of faith in the subsequent consistency of the whole mathematical enterprise!
So in a qualified sense the prime numbers themselves vibrate! More correctly it is the interaction of both the prime and natural numbers that leads to the vibrations corresponding to the Riemann non-trivial zeros. Once again though the starting code is set in an absolute ineffable manner, these vibrations can only become manifest in a phenomenal physical manner (where some degree of separation of both quantitative and qualitative aspects has already taken place).
So the fascinating conclusion that we can draw from this is that manifest physical reality itself represents, at a more fundamental level, vibrations corresponding to number patterns (which are thereby inherent in all physical forms).
Now this might appear ludicrous when we think of numbers with respect to their mere quantitative characteristics. However when we equally recognise the qualitative dimensional characteristics (associated with number) it seems - at least to me - a somewhat obvious conclusion.
And this is the precise nub of the matter. Properly understood the dimensional aspect (i.e. power or exponent) of a number is of a qualitative nature (with respect to its number quantity). However this simple truth is - seemingly - entirely missed in conventional Type 1 appreciation. So when the number a is raised to a corresponding dimensional number b (as power or exponent) i.e. a^b, then - properly understood a and b are quantitative and qualitative with respect to each other.
A simple clue to this truth is given by the fact that when 1 is raised to a dimension that is a rational fraction such as 1/3, a circular number results (i.e. lying on the circle of unit radius in the complex plane). So though 1 and 1/3 are conventionally interpreted as linear quantities (i.e. as points on the real number line) clearly a profound transformation results (from linear to circular form) in relating the base quantity 1 with its dimensional exponent 1/3. So the simple truth is that these numbers - which can indeed be given a quantitative definition in isolation - are actually quantitative and qualitative in relation to each other!
And Type 2 i.e. Holistic Mathematics (which is directly of a qualitative nature) is based on circular rather than linear logical notions!
One important implication of this finding - which I was even dimly aware of as a child - is that one cannot hope to give a coherent explanation of an apparently simple relationship such as the square root of 1 in the absence of Type 2 (qualitative) mathematical understanding!
So remarkably, because of the unquestioned quantitative bias of Conventional Mathematics (which is thereby quite unbalanced) the most obvious problems of mathematical interpretation are persistently overlooked!
Therefore, for example when a number is raised to a whole number dimension (other than 1) in conventional terms a merely reduced quantitative interpretation is given of the result.
Thus 2^3 in Type 1 terms = 8 (i.e. 8^1).
Now by looking at this in a geometrical manner we can easily appreciate that a dimensional transformation of a qualitative nature is equally involved. In other words 2^3 represents cube (3-dimensional) rather than linear (1-dimensional) units.
So, as I have repeatedly expressed, Conventional (Type 1) Mathematics necessarily reduces the qualitative notion of dimension in a merely linear (1-dimensional) fashion.
However, when we begin to appreciate the true qualitative nature of dimension, we perhaps can recognise that physical dimensions - when correctly understood - are intimately related to holistic mathematical notions (of a qualitative Type 2 nature).
I have expressed for some time - see my Integral Science blog - dissatisfaction with the philosophical notion of a string (which has no strict physical meaning).
However once we accept that physical phenomena represent a certain rigid reduction with respect to original vibration patterns of a purely mathematical nature, then we can perhaps appreciate the ultimate nature of strings as the prime number constituents of natural reality with the relationship collectively between strings equally representing the prime number dimensional aspects of this same natural reality.
Reality at its most fundamental level is written in a double binary code (1 and 0) where both quantitative and qualitative aspects interact. This provides a means therefore of potentially encoding all (quantitative) information as is well recognised in the present digital age; however what is not equally recognised is that the same digits when holistically interpreted in a qualitative manner can potentially encode all transformation processes!
Then at the next most fundamental level, reality is written in the prime number code (again with respect to its quantitative and qualitative aspects). And from this code ultimately all phenomenal reality, as we know it, is derived!
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