Tuesday, September 30, 2014

Holistic Synchronous Nature of Number System (1)

The Erdős-Kac Theorem relates to the distribution of the (distinct) prime factors of a number demonstrating that it approaches a perfect normal distribution when the relevant numbers (to which the prime factors relate) are sufficiently large.

So for example if we were in the region of the number system where n > 1020 and were to take in this vicinity a significant sampling of numbers we would find that the number of (distinct) prime factors would vary considerably with the majority close to a central value with higher and lower recorded values symmetrically distributed around the central value in accordance with the normal distribution.

We would equally expect the distribution of H’s with respect to similar sized samples, relating to the tossing of an unbiased coin, to be likewise normally distributed thereby indicating the random nature of the trials involved.

So likewise the Erdős-Kac Theorem demonstrates the random nature with respect to the  occurrence of the distinct prime factors of a number.

However there is also the more familiar way in which we speak about the randomness of the primes which relates to the distribution of the (individual) primes with respect to the overall natural number system.

Thus when we look at these two aspects of prime randomness in the appropriate dynamic interactive manner, we realise that they represent complementary aspects of prime behaviour (with respect to the natural numbers).

So the familiar Type 1 notion entails the (individual) primes in the context of the (collective) natural number system.

However the alternative Type 2 notion entails the complementary notion of a (collective) group of prime factors in the context of a related (individual) natural number.

Also the Type 1 notion entails viewing the primes as base quantities. In relative terms the Type 2 notion relates to the dimensional qualitative aspect of these numbers (as factors).

Notice also how the relationship as between the primes and natural numbers is inverted as we switch from the Type 1 to Type 2 definition! In Type 1 terms the (collective) natural number system is understood in terms of its (individual) prime constituents. In Type 2 terms the primes as a (collective) group of factors are understood in terms of each (individual) natural number!


Thus in truth the relationship the two-way relationship as between the primes and the natural numbers is of a dynamic interactive nature, entailing the complementarity of these two opposite polarities.

Now we have that the notion of randomness itself can only be properly appreciated in a dynamic interactive context where it always implies the opposite notion of order.

Therefore the Type 1 and Type 2 aspects of the primes intimately depend on each other in a holistic synchronous manner.

Now whereas the (Type 1) individual primes are random, we also have the opposite demonstrated characteristic of their overall collective order with respect to the natural numbers (when taken as an entire group) as for example expressed through the prime number theorem.

Likewise whereas the (Type 2) collection of primes is random, we also have the opposite characteristic of individual order (indirectly expressed through the prime roots of 1, where all roots other than 1 are uniquely determined).

Therefore we can validly say that the random nature of the primes in Type 1 terms is intimately related to their corresponding ordered nature in a Type 2 manner; likewise the random nature of the primes in Type 2 terms is intimately related to their corresponding ordered nature in Type 1 fashion.


Thus we can now state this remarkable overriding fact regarding the nature of the number system, which contains the seeds to not only dramatically revolutionise our very appreciation of the nature of Mathematics but likewise all of its related sciences.

And this fact simply relates to the clear recognition of the inherently dynamic nature of the number system with its underlying holistic synchronous basis.

                

Wednesday, September 24, 2014

Using Frequency Formula to Estimate Location of Riemann Zeros

I have remarked before on the stunning accuracy of the formula for calculating the frequency of the Riemann zeros.

Therefore to calculate the frequency of these zeros up to t on the imaginary line (through ½) we use the formula,

t/2π(log t/2π –  1).

I have suggested that in general an even more accurate measurement (in absolute terms) will occur from the addition of 1 to this formula i.e.

t/2π(log t/2π –  1) + 1.

If we use this latter formula to calculate the exact occurrence of just one zero, it occurs at 17.08 (approx) which lies pretty well midway between the first two actual zeros (i.e. 14.13 and 21.02 respectively.

I then went on using the formula to calculate the exact predicted frequency of 2, 3, 4,….10 zeros.

So the table underneath (cols 2 and 3) compares the actual location of the first 10 zeros as against the predicted location. I also show (in cols 4 and 5) the successive deviations of the actual and predicted zeros.


Riemann Zeros
Actual location
Predicted Location
Deviation of Actual Zeros
Deviation of Predicted zeros
       1st
      14.13
     17.08
    
     
       2nd
      21.02
     22.56
     6.89
      5.48
       3rd
      25.01
     27.14
     3.99
      4.58
       4th
      30.43
     31.24
     5.42
      4.10
       5th
      32.94
     35.04
     2.51
      3.80
       6th
      37.59
     38.56
     4.65
      3.52
       7th
      40.92
     41.92
     3.33
      3.36
       8th
      43.33
     45.18
     2.41
      3.26
       9th
      48.01
     48.34
     4.68
      3.16
     10th
      49.77
     51.34
     1.76
      3.00

In one way it is remarkable how this general formula to calculate the frequency of the zeros can be used to estimate the precise value of each zero.

The deviations are also interesting. The deviations with respect to the estimated zeros operate in a smooth consistent fashion with respect to the manner in which they become smaller.

However the corresponding deviations with respect to the actual zeros are much more erratic.

Thus though the average deviation in both cases is roughly similar (3.96 for actual and 3.81 for estimated zeros respectively) considerable variations with respect to the local spread as between any two successive zeros is in evidence.

Monday, September 22, 2014

Interesting Log Relationships (4)

Once again we start with the general expression.

log n – log (n – 1) = 1/n + 1/2n2 + 1/3n3 + 1/4n4 +….

If we let n = – 1, then n – 1 = – 2.

Therefore, log n – log (n – 1) = log (– 1) – log (– 2) = log (– 1/– 2) = log (1/2)

And through the formula expansion,

log (1/2)  = – 1 + 1/2 – 1/3 + 1/4 –…,

= – (1 – 1/2 + 1/3 – 1/4 +…) =  – log 2 = – .693147…

What is interesting here is that we can use the logs of two negative numbers, to derive the well known log of a positive.

Now through Euler’s Identity,

e = – 1, so that log (– 1) = iπ.

And as log (– 1) – log (– 2) = log (1/2),

iπ – log (– 2) = log (1/2),

so that log (– 2) = iπ – log (1/2),

= iπ + log 2.

More generally, we can therefore express the log of any negative number i.e. log (– n), through the complex expression a + it, where a = log n and t = π.


There is a very important point that needs to be made at this point.

Just as in conventional (Type 1) terms we can give the customary analytic interpretation of such mathematical symbols in a quantitative manner, equally in the - as yet - unrecognised (Type 2) terms we can give these symbols a unique holistic interpretation in a qualitative manner.

So the fact that the log of a negative number must be expressed in a complex mathematical fashion (with real and imaginary parts) implies in holistic terms that we properly require here the intermingling of both quantitative interpretation (with respect to the real part) and a corresponding qualitative interpretation (with respect to the imaginary part).

This is a matter over which great confusion presently exists with respect to the standard conventional interpretation of complex logs.

So e = – 1.

Therefore squaring both sides,  

e2iπ = 1

This implies - from the conventional (Type 1) quantitative perspective - that when we multiply the expression on the RHS by e2iπ, that the value remains unchanged as 1.

Therefore,

e2iπ = e4iπ = e6iπ = e8iπ =…..  = 1.

However from the Type 2 perspective it looks very different!

So properly, e2iπ = 11, e4iπ = 12, e6iπ = 13, e8iπ = 14, and so on.

Therefore though the value of these expressions does indeed remain unchanged in a (reduced) quantitative manner according to standard Type 1 interpretation, this value continually changes with respect to the dimensional number involved which - relatively - should be interpreted in a qualitative Type 2 manner.

A coherent interpretation therefore of the nature of the complex behaviour of the logs of negative numbers therefore ultimately requires both analytic (Type 1) and holistic (Type 2) interpretation, which inherently requires a dynamic interactive manner of understanding.

Saturday, September 20, 2014

Interesting Log Relationships (3)

The formula, log n – log (n – 1) = 1/n + 1/2n2 + 1/3n3 + 1/4n4 +…., can be used to generate a simple infinite series expression for every number.

Basically if n is of the form t/(t – 1), then log n – log (n – 1) is of the form,

log t/(t – 1) – log (1/t – 1) = log t.

So, for example to find log 2 we set t = 2.

Therefore log (2/1) – log (1/1) =  log 2.

Thus with n in this case = 2/1,

log 2 = 1/1.2 + 1/2.22 + 1/3.23 + 1/4.24 +…  

So summing the first four terms of the series we obtain .68229…, which already approximates well to the correct answer .693147…

Likewise to find the corresponding expression for log 3 we set t = 3, with n thereby = 3/2.

Thus log 3 = 2/1.3 + 22/2.32 + 23 /3.33 + 24/4.34 +…  

So again by summing the first four terms we obtain 1.03703… as against the correct answer, 1.09861…

This indicates that in this case the answer converges more slowly to the actual result and this will be increasingly the case as n becomes larger!.


Indeed, in principle we can use the same formula i.e..

log n – log (n – 1) = 1/n + 1/2n2 + 1/3n3 + 1/4n4 +…,

to find an unlimited number of infinite expressions for the log of any number.

For example, 4 = 22 and 8 = 23, so we equally express log 2 in terms of any number that is a power of 2 and by extension the log of any number in terms of a number that is a corresponding power of that number..

So log 2 = (log 4)/2 and (log 8)/3 respectively!

Thus these two alternative infinite series expressions for log 2 are,

 (3/1.4 + 32/2.42 + 33 /3.43 + 34/4.44 +… )/2 and

(7/1.8 + 72/2.82 + 73 /3.83 + 74/4.84 +… )/3 respectively.

However, a significant practical problem is that they converge ever more slowly to the actual result.

This problem can however be solved in an interesting manner!  

For example to find a quickly converging series for log 2, we can set

t = 2 raised to a small fractional power!

For example, for t = 21/10, n = t/(t – 1) = 21/10/(21/10 – 1)

Therefore

log 2 = 10{(21/10 – 1)/21/10 + (21/10 – 1)2/2. 22/10 + …….}

= 10(.066967… + .002242… + ….)

= .69209…

So, the result has already converged quite close to the actual result (i.e. .693147…)

However by raising here 2 to a sufficiently small power, we can approximate the log 2 as closely as we wish to the actual result through sole consideration of the first term.

So in general if a is the number whose log we wish to estimate, and 1/t is the power to which it is raised then the approximation = t(a1/t – 1)/a1/t

Therefore for example, when a = 2 and t = 1,000,000 the approximation for log 2 is given as,

1,000,000{(21/1,000,000 – 1)/21/1,000,000} 

= 1,000,000} (1.0000006931474… – 1)/1.0000006931474},

= 1,000,000(.0000006931469…) = .6931469…

So this answer is already correct to 6 significant figures with respect to the actual result.

Indeed as the denominator a/1/t approximates ever more closely to 1 for very large t, the log for any number a can be approximated closely by the even more simple expression,

t(a1/t – 1) where t is sufficiently small.


Again for example, log 145 = 4.97673374242…

So in this case setting t = 1,000,000,000,

a1/t – 1 = 1.0000000049767337548… – 1 = .0000000049767337548…

Therefore log 145 ~ 1,000,000,000(.0000000049767337548…)

= 4.9767337548…

As we can see this approximation is already correct to 7 significant figures.

And we can continually improve this approximation by making t progressively larger.

Tuesday, September 16, 2014

Interesting Log Relationships (2)

In yesterday’s blog entry, I concluded by illustrating that the average spread between primes (to n) is complemented by the average frequency of natural number factors (per unit).

This in fact points to the synchronous nature of the number system where complementary aspects of behaviour dynamically interact with each other.

Put another way, the behaviour with respect to the primes regarding the average spread between each member at various intervals of the number system intimately depends on the corresponding behaviour with respect to the (average) frequency of natural number factors over these same intervals.
Equally, in reverse, the behaviour with respect to frequency of the natural number factors intimately depends on the corresponding behaviour with respect to the average spread as between the primes.

In other words both aspects of number behaviour co-determine each other in a holistic synchronistic manner.

However to properly appreciate the qualitative nature of such synchronistic behaviour, we must inherently view the number system in a dynamic manner (representing again the interaction of complementary opposite poles of behaviour).


Now the harmonic series,

1 + 1/2 + 1/3 + 1/4 + …..1/n  = log n + γ (where γ is the Euler-Mascheroni constant = .5772… approx).

Once again,

log n – log (n – 1) = 1/n + 1/2n2 + 1/3n3 + 1/4n4 +……

Therefore when n = 2,

log 2 – log 1 = 1/2 + (1/2n2 + 1/3n3 + 1/4n4+…… ) where again n = 2.

Likewise,

log 3 – log 2 = 1/3 + (1/2n2 + 1/3n3 + 1/4n4 +……) where again  n = 3

and

log 4 – log 3 = 1/4 + (1/2n2 + 1/3n3 + 1/4n4 +……) where again n = 4

and continuing on in this fashion, finally,

log n – log (n – 1) = 1/n + (1/2n2 + 1/3n3 + 1/4n4 +……) where again  n = n.

Therefore summing up terms on both LHS and RHS

log n – log 1 = 1/2 + 1/3 + 1/4 + …… 1/n + ∑(1/2n2 + 1/3n3 + 1/4n4 +……) where the value of n is taken from 2 to n.  

Therefore because log 1 (in real terms) = 0, then

log n = log n + γ – 1 + ∑(1/2n2 + 1/3n3 + 1/4n4 +…… ).

Therefore γ = 1 – ∑(1/2n2 + 1/3n3 + 1/4n4 +……), again summed from 2 to n.

So this provides one interesting way of expressing the value of the Euler-Mascheroni constant!


Interestingly when n = 1, the Riemann Zeta Function i.e. ζ(1) (which results in the harmonic series) and the expression for log n – log (n – 1) are identical.

Indeed γ can equally be given a fascinating expression in terms of the Riemann Zeta Function so that,

λ ζ(2)/2 - ζ(3)/3 + ζ(4)/4 - ζ(5)/5 +….. (when summed to a finite number of terms with approximation improving as the number of terms increases).

This would imply therefore that
  
log n → ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 - …..

Alternatively,

λ → ζ(1) – {ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +…..} again when summed to a finite n, with approximation improving as n improves.










     

Monday, September 15, 2014

Interesting Log Relationships (1)

As is well known the average spread or gap between primes (to n) is measured approximately by log n.

For example when n = 100, log n = 4.605…, so we would expect the average gap between primes in the region of 100 to lie somewhere between 4 and 5 (approximately).

This relative measurement will then steadily improve in accuracy for larger values of n.

Therefore the change in the spread between primes as the natural number increases by 1 to n is given as log n – log (n – 1) = log n/(n – 1).

This difference of logs is given by the infinite expression,

i.e. log n – log (n – 1) = 1/n + 1/2n2 + 1/3n3 + 1/4n4 +……


For large n this entails that log n – log (n – 1) is approximated very accurately by 1/n.

For example if n = 1,000, then log n – log (n – 1) = log 1,000 – log 999

= 6.9077552… –  6.9067547… = .0010005…

And this result - for a still comparatively low value of n - is already very close to the reciprocal of n, i.e. 1/1000 = .001.

Indeed if we take for greater accuracy the midpoint of the two numbers 999 and 1000, i.e. 999.5, then the result from subtracting the two log expressions is extremely close to the reciprocal of 999.5, i.e. .0010000500025…

What we have illustrated here stands here as perhaps the most remarkable - yet simple - example of the nature of prime behaviour in that the change in the average spread (or gap) as between successive prime numbers at n is intimately related to the corresponding reciprocal of n.

In truth, the average spread between primes is better approximated by log n – 1 and this distinction would be indeed significant over the lower ranges of n!

For example there are 78,498 primes less than 1,000,000.

Now using log n as the estimate for the average spread as between primes, this would indicate using the formula n/log n, a total of 72,382 primes up to 1,000,000.

However using log n – 1 as the average spread between primes this would give, using the corresponding formula, n/ log n – 1, the much more accurate total of 78,030 primes (to 1,000,000).

However for very large n, the distinction as between log n and log n – 1 becomes increasingly less significant in relative terms.


What is not commonly realised is that log n has an equally important complementary significance with respect to the natural factors of a number.

Once, again these natural factors are defined solely with respect to the composite natural numbers and relate to all the factors of such a number (except 1).

So for example, 12 contains 5 natural factors i.e. 2, 3, 4, 6 and 12!

Now, log n - though once again log n – 1 is the better estimate for smaller values of n - measures the average number of natural factors per unit to n.

Therefore, for example when n = 100, we would say that the average number of factors per unit = 4.605…

This would therefore suggest a cumulative total of 461 natural factors to 100.

However, because n is still very small, as we have seen, log n – 1, would - relatively - provide the better estimate.

As log 100 – 1= 3.605…,, this would give a cumulative estimate of factors up to 100 of 361.

Indeed this is very close to the actual cumulative total of factors to 100, which is 357!    

Therefore we can equally say that the change in the average frequency of natural factors per unit as n increases by 1 is given by the reciprocal of n = 1/n.

So once again, as we have seen, when for example, n = 1,000, the change in the average gap between primes (per unit) is approximated by 1/1,000, likewise when n = 1,000, the average change in the frequency of natural factors (per unit) is likewise approximated by 1/1,000.