Conventional physical accounts of the origin of the universe centre on the Big Bang where in an incredibly short period all the fundamental constituents of matter were formed.

However I have argued that the notion of ultimate particles has no strict meaning (in a manifest form) and that matter particles themselves represent more fundamental number interactions with respect to their quantitative and qualitative aspects.

So ultimately all notions of quantity and quality are of a strict mathematical nature. And it is in the dynamic interaction of these two aspects that physical reality - as we know it - arises.

Thus if the Big Bang is to have any true meaning it most fundamentally relates to the manner in which number - again with respect to its specific (quantitative) and holistic (qualitative) aspects - itself arises.

Now of particular interest here would be the primes (and their dynamic relationship with the natural numbers).

As I have demonstrated the manner to which both unfold corresponds to a dynamic interactive process based on an initial binary logical system.

So one again in this system the linear (1) is based on unambiguous positing (+).

The circular (0) is then based on both positing and negating.

So 1 - 1 = 0 (in this system).

The initial prime number is 2 (which holistically represents duality!).

Subsequent (composite) natural numbers are uniquely determined though multiplication of prexisting primes. Subsequent primes are uniquely determined from already (composite) natural numbers through subtraction of 1.

So rather that statically existing, prime and natural numbers are formed through a dynamic interactive process designed to preserve the unique identity of each prime and also the collective interdependence of all primes (with the natural numbers).

As it is a phenomenal process taking place in space and time the primes can never be fully reconciled with natural numbers in this relative manner. However the approximation to the initial state (where quantitative and qualitative aspects are identical) would occur ever closer to the moment of the Number Bang. Before quantitative or qualitative aspects of creation had yet emerged by definition the perfect reconciliation of the number system existed in an ineffable manner.

From one valid perspective, evolution could be looked on as the attempt to fully "know" the nature of this initial ineffable number state.

But once again such knowledge is ultimately also of an ineffable nature. Thus in pure spiritual union where all physical phenomena have been made transparent, one can approach that state in knowledge of the true nature of the primes.

For such knowledge of the mystery inherent in the primes is inseparable from the true nature of existence!

## Saturday, December 31, 2011

## Friday, December 30, 2011

### Changing Our Ideas on the Primes

I have continually asserted the fact that the way we fundamentally look at the primes is very misleading.

Arising from the linear (1-dimensional) nature of Conventional Mathematics, the primes are viewed as the basic (independent) building blocs of the natural number system.

However even momentary reflection on the matter would immediately lead to the realisation that we cannot even begin to think of the primes in the absence of the natural numbers.

For example as soon as we try to rank primes we automatically are required to use the natural numbers in an ordinal sense. So 2 is the 1st, 3 the 2nd, 5 the 3rd, 7 the 4th prime and so on.

But the ordinal natural numbers in turn implies the cardinal use of these numbers. So we cannot give 1st, 2nd, 3rd for example a meaning in the absence of the cardinal numbers 1, 2, 3!

Thus the prime and the natural numbers are mutually interdependent.

This requires adopting the radical view that just as the natural numbers (from one perspective) are determined by the primes, that the primes in turn are determined by the natural numbers.

Now the clue to this new realisation comes from recognition that there are in fact two key logical systems. The linear logical system based on unambiguous either/or distinctions (on which Conventional Mathematics depends) can be represented as + 1.

However the alternative circular logical system based on relative both/and can be represented as 1 - 1 = 0.

Clearly before phenomena can unfold, both the linear and circular systems are identical i.e. as undifferentiated form (which equally is emptiness).

Now the conventional notion that natural numbers are derived from the primes is based on the linear logical approach.

However the reverse view that the primes are derived from the natural numbers springs from appreciation of the circular approach.

Now initially the circular is represented as 1 - 1 = 0.

However with the emergence of natural numbers it is represented as n - 1 (where n is a natural number).

The point about this is that all new primes can be represented uniquely by a natural number (that has been derived from earlier primes - 1).

For example 29 (which is a new prime) can be uniquely derived as 30 - 1.

30 in this case is the composite result of multiplying the earlier primes 2, 3 and 5.

Thus 29 can be uniquely represented as (2 * 3 * 5) - 1.

So the number system starts with 1 and 0 with 0 itself represented as 1 - 1.

Now the origin of the next number 2 (leading to the birth of duality) arises from portraying the circular system in linear terms. Thus rather that preserving the dynamic complementarity of + and - signs, both are represented as +.

Thus instead of 1 - 1 = 0, we now have 1 + 1 = 2.

All subsequent natural and prime numbers are ultimately derived from 2 (and 1).

So 3 the next prime number = (2 * 2) - 1 = 3.

Then the next prime number 5 arises from (2 * 3) - 1 = 5.

So remarkably we have two systems here!

The first (linear) system is based on multiplication of primes (to derive the natural numbers).

The second (circular) system is based on subtraction of 1 (from the composite natural number result of multiplying primes) to uniquely define new prime numbers.

So in actual reality (and experience) both systems are involved. The natural numbers and primes are built up in conjunction with each other in a dynamic interactive manner.

In a very valid sense neither the primes nor natural numbers preexist this dynamic interactive process that takes place in space and time.

Thus the prime numbers and the natural can never be fully reconciled with each other in a formal phenomenal procedure. Such reconciliation takes place in an ineffable manner (as the secret code pre-existing phenomenal creation) and equally the ultimate spiritual realisation of the true mystery of the primes (and natural numbers).

Incidentally the true circular nature of the number 0 is revealed by the fact that is invariant with respect to either positive or negative signs. So + 0 = - 0.

What this really means is that 0 properly embraces both positive and negative signs in a circular manner! And the very symbol we use for 0 signifies this circular nature (just as the symbol we use to represent 1 likewise symbolises its linear nature)!

Arising from the linear (1-dimensional) nature of Conventional Mathematics, the primes are viewed as the basic (independent) building blocs of the natural number system.

However even momentary reflection on the matter would immediately lead to the realisation that we cannot even begin to think of the primes in the absence of the natural numbers.

For example as soon as we try to rank primes we automatically are required to use the natural numbers in an ordinal sense. So 2 is the 1st, 3 the 2nd, 5 the 3rd, 7 the 4th prime and so on.

But the ordinal natural numbers in turn implies the cardinal use of these numbers. So we cannot give 1st, 2nd, 3rd for example a meaning in the absence of the cardinal numbers 1, 2, 3!

Thus the prime and the natural numbers are mutually interdependent.

This requires adopting the radical view that just as the natural numbers (from one perspective) are determined by the primes, that the primes in turn are determined by the natural numbers.

Now the clue to this new realisation comes from recognition that there are in fact two key logical systems. The linear logical system based on unambiguous either/or distinctions (on which Conventional Mathematics depends) can be represented as + 1.

However the alternative circular logical system based on relative both/and can be represented as 1 - 1 = 0.

Clearly before phenomena can unfold, both the linear and circular systems are identical i.e. as undifferentiated form (which equally is emptiness).

Now the conventional notion that natural numbers are derived from the primes is based on the linear logical approach.

However the reverse view that the primes are derived from the natural numbers springs from appreciation of the circular approach.

Now initially the circular is represented as 1 - 1 = 0.

However with the emergence of natural numbers it is represented as n - 1 (where n is a natural number).

The point about this is that all new primes can be represented uniquely by a natural number (that has been derived from earlier primes - 1).

For example 29 (which is a new prime) can be uniquely derived as 30 - 1.

30 in this case is the composite result of multiplying the earlier primes 2, 3 and 5.

Thus 29 can be uniquely represented as (2 * 3 * 5) - 1.

So the number system starts with 1 and 0 with 0 itself represented as 1 - 1.

Now the origin of the next number 2 (leading to the birth of duality) arises from portraying the circular system in linear terms. Thus rather that preserving the dynamic complementarity of + and - signs, both are represented as +.

Thus instead of 1 - 1 = 0, we now have 1 + 1 = 2.

All subsequent natural and prime numbers are ultimately derived from 2 (and 1).

So 3 the next prime number = (2 * 2) - 1 = 3.

Then the next prime number 5 arises from (2 * 3) - 1 = 5.

So remarkably we have two systems here!

The first (linear) system is based on multiplication of primes (to derive the natural numbers).

The second (circular) system is based on subtraction of 1 (from the composite natural number result of multiplying primes) to uniquely define new prime numbers.

So in actual reality (and experience) both systems are involved. The natural numbers and primes are built up in conjunction with each other in a dynamic interactive manner.

In a very valid sense neither the primes nor natural numbers preexist this dynamic interactive process that takes place in space and time.

Thus the prime numbers and the natural can never be fully reconciled with each other in a formal phenomenal procedure. Such reconciliation takes place in an ineffable manner (as the secret code pre-existing phenomenal creation) and equally the ultimate spiritual realisation of the true mystery of the primes (and natural numbers).

Incidentally the true circular nature of the number 0 is revealed by the fact that is invariant with respect to either positive or negative signs. So + 0 = - 0.

What this really means is that 0 properly embraces both positive and negative signs in a circular manner! And the very symbol we use for 0 signifies this circular nature (just as the symbol we use to represent 1 likewise symbolises its linear nature)!

## Tuesday, December 27, 2011

### Vibrating Primes

The primes seemingly give rise to a dynamic vibrating system that is of a physical nature.

Once again there are quantitative (actual) and qualitative (holistic) aspects to the primes, which ultimately are identical in an ineffable formless state (that precedes physical existence).

However as soon as the aspects separate the primes take on a physical existence and become manifest in phenomena with both quantitative and qualitative aspects (that are to a degree separated in space and time).

Physicists now speak of strings as being the ultimate particles. So in this worldview the strings vibrate giving rise to the physical particles of nature.

Now the weakness of this view is that strings are given a merely actual existence. This leads to the mistaken - and highly reductionist - view that the fundamental constituents of matter i.e. strings) are of a uniform nature devoid of any distinctive qualitative features!

Some time ago I had already formed the view that strings must equally possess unique holistic dimensional properties (conforming to a distinctive circular logical system) and that through the interaction of both actual and holistic aspects that physical nature as we know it arises.

However I have now come to realise that ultimately what are termed "strings" relate directly to the prime numbers (with respect to both their quantitative and qualitative characteristics).

Thus what we call physical nature arises directly from the dynamic interaction of the actual and holistic nature of prime numbers (that are utterly unique yet collectively interdependent with each other).

Strictly this vibration of prime numbers is already inherent in fundamental natural particles. In one way this vibration can be seen as an attempt to get back to that ineffable state (where form does not yet exist). From an equally valid perspective it can be seen as the attempt to set the evolution of nature in process where the true understanding of its underlying prime nature can be ultimately attained once again in an ineffable formless manner.

So as we come ever closer to the original state of matter, we equally come ever closer to approaching the secret code contained in the prime numbers which determines the whole subsequent course of evolution.

There are of course no ultimate particles in nature (that are detectable). Rather beyond what is detectable are extremely dynamic physical interactions of an increasingly transient and elusive nature. And these interactions approximate ever more closely to the number code (that perfectly reconciles quantitative and qualitative aspects of the primes) which ultimately is of an ineffable nature.

Once again there are quantitative (actual) and qualitative (holistic) aspects to the primes, which ultimately are identical in an ineffable formless state (that precedes physical existence).

However as soon as the aspects separate the primes take on a physical existence and become manifest in phenomena with both quantitative and qualitative aspects (that are to a degree separated in space and time).

Physicists now speak of strings as being the ultimate particles. So in this worldview the strings vibrate giving rise to the physical particles of nature.

Now the weakness of this view is that strings are given a merely actual existence. This leads to the mistaken - and highly reductionist - view that the fundamental constituents of matter i.e. strings) are of a uniform nature devoid of any distinctive qualitative features!

Some time ago I had already formed the view that strings must equally possess unique holistic dimensional properties (conforming to a distinctive circular logical system) and that through the interaction of both actual and holistic aspects that physical nature as we know it arises.

However I have now come to realise that ultimately what are termed "strings" relate directly to the prime numbers (with respect to both their quantitative and qualitative characteristics).

Thus what we call physical nature arises directly from the dynamic interaction of the actual and holistic nature of prime numbers (that are utterly unique yet collectively interdependent with each other).

Strictly this vibration of prime numbers is already inherent in fundamental natural particles. In one way this vibration can be seen as an attempt to get back to that ineffable state (where form does not yet exist). From an equally valid perspective it can be seen as the attempt to set the evolution of nature in process where the true understanding of its underlying prime nature can be ultimately attained once again in an ineffable formless manner.

So as we come ever closer to the original state of matter, we equally come ever closer to approaching the secret code contained in the prime numbers which determines the whole subsequent course of evolution.

There are of course no ultimate particles in nature (that are detectable). Rather beyond what is detectable are extremely dynamic physical interactions of an increasingly transient and elusive nature. And these interactions approximate ever more closely to the number code (that perfectly reconciles quantitative and qualitative aspects of the primes) which ultimately is of an ineffable nature.

## Wednesday, December 21, 2011

### Sum of Reciprocals of Primes

As is well known the sum of the terms in the harmonic series

1 + 1/2 + 1/3 + 1/4 +.... ~ ln n + γ (where γ = the Euler-Mascheroni constant =.5772..)

It is fascinating therefore that the sum of reciprocals of primes

1/2 + 1/3 + 1/5 + 1/7 + ..... ~ ln ln n + B (where B = Merten's constant = .261497..)

This would of course suggest that the sum of this series diverges for large n!

However just as the harmonic series can be used to calculate the spread as between cardinal prime numbers, likewise this latter series can be used to calculate the spread as between ordinal prime numbers.

In other words all prime numbers can be linked with the ordinal set of natural numbers.

So 2 is the 1st, 3 the 2nd, 5 the 3rd, 7 the 4th prime respectively.

So if we now order these primes in an ordinal prime fashion, then both 3 and 5 are prime (i.e. as the 2nd and 3rd primes).

We could then reorder these surviving primes in an natural number ordinal fashion before selecting once again those surviving numbers that are prime in an ordinal fashion.

Let us take all the primes up to 31 to illustrate,

(1) 2, (2) 3, (3) 5, (4) 7, (5) 11, (6) 13, (7) 17, (8) 19, (9) 23, (10) 29, (11) 31.

These these starting cardinal primes are listed in natural number ordinal fashion (in brackets). We will refer to these as Order 1 Primes. So all prime numbers are Order 1 primes.

Then if we extract the primes (whose ordinal numbers are also prime), we are left with 3, 5 11, 17 and 31.

If we now again rank these ordinally in natural number fashion we have

(1) 3, (2) 5, (3) 11, (4) 17 and (5) 31.

We can refer to this smaller group as Order 2 Primes.

Once again we can then extract only those primes that have a prime number ordinal ranking i.e. 5, 11 and 31.

Then shifting to ordinal natural number ranking we have a new - even smaller - set of surviving primes,

(1) 5, (2) 11 and (3) 31.

We can refer to these then as Order 3 Primes.

Then once more extracting those remaining with an ordinal prime ranking we are left with 11 and 31

So giving natural number ordinal rankings we have,

(1) 11 and (2) 31

These are Order 4 Primes.

Finally extracting the one remaining prime with an ordinal prime ranking we are left with 31.

Finally ranking this as (1) 31, this qualifies as an Order 5 Prime.

It is no accident that 31 corresponds to the Mersenne prime 2^n - 1 (where n = 5).

Indeed if we were to continue up to 127 for example, 127 would then qualify as the one remaining Order 7 Prime. And 127 is the Mersenne prime (where n = 7).

I have suggested at various times that perhaps we could guarantee the generation of Mersenne primes by starting with 2 and then proceeding through switching in an orderly fashion as between quantitative (base) and qualitative (dimensional) use of prime numbers.

So 2^2 - 1 = 3 (as base number).

Then substituting 3 as dimensional number we have,

2^3 - 1 = 7 (which is prime).

Once again substituting 7 as dimension we have,

2^7 - 1 = 127 (which is prime).

Then substituting 127 as dimension we have,

2^127 - 1 = 170141183460469231731687303715884105727 (which is prime).

It is tempting to argue that by using this number as exponent of 2 that we can generate a new Mersenne prime that is incomparably larger than any yet discovered!

Now going back to the harmonic series and sum of the reciprocals of primes.

Once again the sum of the harmonic series for large n approximates to ln n.

The sum of the reciprocals of primes for large n approximates to ln ln n.

Now the prime numbers used as denominators in this series are Order 1 Primes.

It is possible therefore to extend this result for Order 2, Order 3, Order 4 primes etc.

For example ultimately the sum of reciprocals of Order 2 Primes should approximate (for sufficiently large n) to Ln Ln Ln n, Order 3 to Ln Ln Ln Ln n, Order 4 to Ln Ln Ln Ln Ln n etc.

This would suggest that no matter how high the Order of Primes involved that the sum of the series of its reciprocal terms would diverge (for sufficiently large n).

Also by this reckoning the harmonic series could be interpreted as the sum of Reciprocals of Order 0 Primes.

So the natural numbers are prime numbers of Order 0!

What simply this means in effect is that the ordinal ranking of the complete set of primes (i.e. Order 1 Primes) is given by the natural numbers!

As for the spread as between primes, once again for Order 1 Primes the answer approximates to log n.

Clearly the spread will grow for higher Order Primes.

So we can postulate that the number of Order 1 Primes up to 1,000,000 = n/ln n = n1 = 72,283 (approx).

Therefore the number of Order 2 Primes up to 1,000,000 approximates n1/ln n1 = 6469 (approx).

Thus the average gap as between Order 2 Primes approximates n/{n1/ln n1} = 1,000,000/6469.

Therefore in the region of 1,000,000 we would expect the average gap as between Order 2 Primes to approximate 154.6.

Now because n is still of a relatively small magnitude, the actual number of Order 2 Primes would differ significantly from this estimate. However the approximation of estimated to actual would continue to improve (in relative terms) as n increases.

The upshot of what we are doing here is that the prime and natural numbers are in fact completely interdependent with each other.

From one perspective the (individual) natural numbers are derived from the primes; however equally from the complementary perspective, the (general) distribution of the primes is derived from the natural numbers.

1 + 1/2 + 1/3 + 1/4 +.... ~ ln n + γ (where γ = the Euler-Mascheroni constant =.5772..)

It is fascinating therefore that the sum of reciprocals of primes

1/2 + 1/3 + 1/5 + 1/7 + ..... ~ ln ln n + B (where B = Merten's constant = .261497..)

This would of course suggest that the sum of this series diverges for large n!

However just as the harmonic series can be used to calculate the spread as between cardinal prime numbers, likewise this latter series can be used to calculate the spread as between ordinal prime numbers.

In other words all prime numbers can be linked with the ordinal set of natural numbers.

So 2 is the 1st, 3 the 2nd, 5 the 3rd, 7 the 4th prime respectively.

So if we now order these primes in an ordinal prime fashion, then both 3 and 5 are prime (i.e. as the 2nd and 3rd primes).

We could then reorder these surviving primes in an natural number ordinal fashion before selecting once again those surviving numbers that are prime in an ordinal fashion.

Let us take all the primes up to 31 to illustrate,

(1) 2, (2) 3, (3) 5, (4) 7, (5) 11, (6) 13, (7) 17, (8) 19, (9) 23, (10) 29, (11) 31.

These these starting cardinal primes are listed in natural number ordinal fashion (in brackets). We will refer to these as Order 1 Primes. So all prime numbers are Order 1 primes.

Then if we extract the primes (whose ordinal numbers are also prime), we are left with 3, 5 11, 17 and 31.

If we now again rank these ordinally in natural number fashion we have

(1) 3, (2) 5, (3) 11, (4) 17 and (5) 31.

We can refer to this smaller group as Order 2 Primes.

Once again we can then extract only those primes that have a prime number ordinal ranking i.e. 5, 11 and 31.

Then shifting to ordinal natural number ranking we have a new - even smaller - set of surviving primes,

(1) 5, (2) 11 and (3) 31.

We can refer to these then as Order 3 Primes.

Then once more extracting those remaining with an ordinal prime ranking we are left with 11 and 31

So giving natural number ordinal rankings we have,

(1) 11 and (2) 31

These are Order 4 Primes.

Finally extracting the one remaining prime with an ordinal prime ranking we are left with 31.

Finally ranking this as (1) 31, this qualifies as an Order 5 Prime.

It is no accident that 31 corresponds to the Mersenne prime 2^n - 1 (where n = 5).

Indeed if we were to continue up to 127 for example, 127 would then qualify as the one remaining Order 7 Prime. And 127 is the Mersenne prime (where n = 7).

I have suggested at various times that perhaps we could guarantee the generation of Mersenne primes by starting with 2 and then proceeding through switching in an orderly fashion as between quantitative (base) and qualitative (dimensional) use of prime numbers.

So 2^2 - 1 = 3 (as base number).

Then substituting 3 as dimensional number we have,

2^3 - 1 = 7 (which is prime).

Once again substituting 7 as dimension we have,

2^7 - 1 = 127 (which is prime).

Then substituting 127 as dimension we have,

2^127 - 1 = 170141183460469231731687303715884105727 (which is prime).

It is tempting to argue that by using this number as exponent of 2 that we can generate a new Mersenne prime that is incomparably larger than any yet discovered!

Now going back to the harmonic series and sum of the reciprocals of primes.

Once again the sum of the harmonic series for large n approximates to ln n.

The sum of the reciprocals of primes for large n approximates to ln ln n.

Now the prime numbers used as denominators in this series are Order 1 Primes.

It is possible therefore to extend this result for Order 2, Order 3, Order 4 primes etc.

For example ultimately the sum of reciprocals of Order 2 Primes should approximate (for sufficiently large n) to Ln Ln Ln n, Order 3 to Ln Ln Ln Ln n, Order 4 to Ln Ln Ln Ln Ln n etc.

This would suggest that no matter how high the Order of Primes involved that the sum of the series of its reciprocal terms would diverge (for sufficiently large n).

Also by this reckoning the harmonic series could be interpreted as the sum of Reciprocals of Order 0 Primes.

So the natural numbers are prime numbers of Order 0!

What simply this means in effect is that the ordinal ranking of the complete set of primes (i.e. Order 1 Primes) is given by the natural numbers!

As for the spread as between primes, once again for Order 1 Primes the answer approximates to log n.

Clearly the spread will grow for higher Order Primes.

So we can postulate that the number of Order 1 Primes up to 1,000,000 = n/ln n = n1 = 72,283 (approx).

Therefore the number of Order 2 Primes up to 1,000,000 approximates n1/ln n1 = 6469 (approx).

Thus the average gap as between Order 2 Primes approximates n/{n1/ln n1} = 1,000,000/6469.

Therefore in the region of 1,000,000 we would expect the average gap as between Order 2 Primes to approximate 154.6.

Now because n is still of a relatively small magnitude, the actual number of Order 2 Primes would differ significantly from this estimate. However the approximation of estimated to actual would continue to improve (in relative terms) as n increases.

The upshot of what we are doing here is that the prime and natural numbers are in fact completely interdependent with each other.

From one perspective the (individual) natural numbers are derived from the primes; however equally from the complementary perspective, the (general) distribution of the primes is derived from the natural numbers.

### Spiral Waves

I have referred before to Matthew Watkin's book "The Mystery of the Prime Numbers".

One of the features that I especially like about this book is that he successfully converts the standard natural log notion (which has such an important bearing on the general distribution of the primes) into the much more expressive form of an equiangular spiral.

Now in geometric terms equiangular spirals are very suggestive as they combine both linear and circular notions in a systematic ordered manner.

Indeed at the two extremes of such spirals we get - what he refers to as degenerative spirals - of both the straight line and the circle.

Matthew also extends this notion of spirals into the treatment of the famous deviations in the Riemann Prime Counting Function which are intimately related in turn to the non-trivial zeros of the Zeta Function!

I look forward very much to an extended treatment of these "spiral zeros" in the second volume.

It is also worth noting that considerable attention has been given to the Prime Spiral (Ulam's Spiral) where when natural numbers are entered on a grid in spiral fashion that the prime numbers then tend to fall along diagonal lines through the spiral! However my intention here is to focus on the qualitative significance of the relationship of prime numbers to spiral wave forms!

Just as linear and circular notions have a well defined quantitative meaning in conventional (Type 1) mathematical terms, equally they have a well defined qualitative meaning in holistic (Type 2) mathematical terms. However this latter type of interpretation is totally ignored by the mathematics profession.

Thus I would strongly contend that it is the absence of this vital qualitative dimension that is preventing recognition of the true nature of prime numbers.

In other words prime numbers combine both specific (independent) aspects in their individual nature with holistic (interdependent) aspects in their overall distribution.

Actual experience of mathematical reality equally combines specific and holistic elements through the interaction of (conscious) reason and (unconscious) intuition. However once again in conventional terms the qualitative (intuitive) aspect is reduced in a quantitative (rational) manner.

Proper understanding of the Riemann Hypothesis thereby requires both linear (either/or) logic based on the clear separation of polar opposites such as external/ internal and circular (both/and) logic based on the corresponding complementarity of such opposites.

Indeed ultimately the Riemann Hypothesis relates to the vital condition necessary for the consistent relationship of both types of logic!

One of the features that I especially like about this book is that he successfully converts the standard natural log notion (which has such an important bearing on the general distribution of the primes) into the much more expressive form of an equiangular spiral.

Now in geometric terms equiangular spirals are very suggestive as they combine both linear and circular notions in a systematic ordered manner.

Indeed at the two extremes of such spirals we get - what he refers to as degenerative spirals - of both the straight line and the circle.

Matthew also extends this notion of spirals into the treatment of the famous deviations in the Riemann Prime Counting Function which are intimately related in turn to the non-trivial zeros of the Zeta Function!

I look forward very much to an extended treatment of these "spiral zeros" in the second volume.

It is also worth noting that considerable attention has been given to the Prime Spiral (Ulam's Spiral) where when natural numbers are entered on a grid in spiral fashion that the prime numbers then tend to fall along diagonal lines through the spiral! However my intention here is to focus on the qualitative significance of the relationship of prime numbers to spiral wave forms!

Just as linear and circular notions have a well defined quantitative meaning in conventional (Type 1) mathematical terms, equally they have a well defined qualitative meaning in holistic (Type 2) mathematical terms. However this latter type of interpretation is totally ignored by the mathematics profession.

Thus I would strongly contend that it is the absence of this vital qualitative dimension that is preventing recognition of the true nature of prime numbers.

In other words prime numbers combine both specific (independent) aspects in their individual nature with holistic (interdependent) aspects in their overall distribution.

Actual experience of mathematical reality equally combines specific and holistic elements through the interaction of (conscious) reason and (unconscious) intuition. However once again in conventional terms the qualitative (intuitive) aspect is reduced in a quantitative (rational) manner.

Proper understanding of the Riemann Hypothesis thereby requires both linear (either/or) logic based on the clear separation of polar opposites such as external/ internal and circular (both/and) logic based on the corresponding complementarity of such opposites.

Indeed ultimately the Riemann Hypothesis relates to the vital condition necessary for the consistent relationship of both types of logic!

## Sunday, December 18, 2011

### Riemann's Zero

As is well known the Riemann Hypothesis amounts to the statement all the non-trivial zeros of the Riemann Zeta Function lie on the real line whose value = 1/2.

Now in conventional terms, mathematicians have been trying to understand this problem from a merely quantitative perspective.

My persistent point however is that Mathematics equally contains an important (largely unrecognised) qualitative aspect.

Furthermore as the Riemann Hypothesis properly relates to the ultimate reconciliation of both the quantitative and qualitative aspects of mathematical understanding, its significance cannot be appreciated in a merely (reduced) quantitative manner.

Indeed ultimately the Riemann Hypothesis points to the relationship as between dual and nondual meaning.

Imagine a circle that is drawn with unit radius. Now in conventional terms the length of the line diameter of this circle = 2 units. So the midpoint of this line at the centre of the circle from linear perspective divides the line into two equal parts. This point lies exactly halfway (1/2) on the total line.

However if we look at this midpoint from a circular perspective we would represent it as 0. In other words if the radius to the right is + 1, then the radius in the opposite direction is - 1.

Now this behaviour has a direct qualitative significance. Linear understanding is inherently dualistic (which represents the holistic meaning of 2). So the midpoint of the circle at 1 unit represents thereby half of the total line.

However to understand the measurement of the line from the centre of the circle in complementary positive and negative directions we require - literally - nondual insight (so that dual notions are rendered paradoxical). So the midpoint from this perspective is 0.

Properly understanding the Riemann Hypothesis requires both dual and nondual understanding. Indeed it points to the ultimate state where both dual and nondual notions (or alternatively quantitative and qualitative notions) of number are identical. And this state is utterly mysterious and thereby cannot be grasped in a phenomenal manner.

Thus the mysterious order governing the nature of the prime numbers is already inherent in all number behaviour representing a non-phenomenal reality (where qualitative cannot be distinguished from quantitative notions).

The attempt to prove the Riemann Hypothesis in a merely (reduced) quantitative manner is thereby utterly futile.

The significance of the Riemann Hypothesis is ultimately of a truly breath taking order in that - properly understood - phenomenal reality as we know it would not be even possible if it did not hold.

In other words underlying all of visible reality is reality is a secret code ensuring a perfect harmony of the prime numbers with respect to their specific (quantitative) and holistic (qualitative) interaction.

It is this harmony that enables all subsequent phenomenal events (representing varying interactions of this original prime number number code) to unfold.

This once again suggests that ultimately the underlying nature of reality - insofar as it can be phenomenally investigated - is purely mathematical!

In other words at the deepest level, phenomena represent the interaction of a fundamental mathematical code that governs the subsequent behaviour of all natural events. However though we can only come to knowledge of this code through phenomena, its nature - and indeed in reverse fashion what we know as nature - is ultimately ineffable.

So at some stage, Physics will have to abandon the quest for the ultimate particles before arriving at a purely mathematical appreciation of reality that underlies all manifestations of such particles.

Indeed properly understood it has already arrived at this point. As I would see it, string theory represents an elaborate fiction that the ultimate physical particles are strings. Strictly speaking however these have no manifest physical reality but really operate as the vehicle of ever purer mathematical notions.

However my key point all along is that the very scope of Mathematics needs to be radically extended to explicitly include both its qualitative as well as quantitative aspect. In this light all particles ultimately emerge as the dynamic interaction of a prime number mathematical code that is designed to preserve the perfect harmony of both its quantitative and qualitative aspects.

Though this point is largely lost because of the reductionist nature of current scientific understanding, the truly great wonder of reality is how both the finite and infinite - though utterly distinct - yet successfully coexist with each other at all levels of understanding.

Thus before we can even for example engage in conventional mathematical activity, we must already presume this meaningful correspondence of finite and infinite. Indeed as I have frequently pointed out it underlines the very notion of mathematical proof!

Looked at another way, the Riemann Hypothesis is the necessary condition for such correspondence to exist. So it represents in fact a massive act of faith in the subsequent meaning of the whole mathematical enterprise.

As the Riemann Hypothesis thereby already underlies conventional mathematical proof (as the starting condition for its meaningful interpretation), The Riemann Hypothesis cannot therefore be proven (in conventional terms).

However far from this representing a defeat, true realisation of this fact has the capacity to open up appreciation of a greatly enlarged scope for Mathematics where every number, symbol, relationship has a unique qualitative - as well as quantitative - significance. And with this will come an enormous enrichment of the true nature of both Mathematics and Science.

Now in conventional terms, mathematicians have been trying to understand this problem from a merely quantitative perspective.

My persistent point however is that Mathematics equally contains an important (largely unrecognised) qualitative aspect.

Furthermore as the Riemann Hypothesis properly relates to the ultimate reconciliation of both the quantitative and qualitative aspects of mathematical understanding, its significance cannot be appreciated in a merely (reduced) quantitative manner.

Indeed ultimately the Riemann Hypothesis points to the relationship as between dual and nondual meaning.

Imagine a circle that is drawn with unit radius. Now in conventional terms the length of the line diameter of this circle = 2 units. So the midpoint of this line at the centre of the circle from linear perspective divides the line into two equal parts. This point lies exactly halfway (1/2) on the total line.

However if we look at this midpoint from a circular perspective we would represent it as 0. In other words if the radius to the right is + 1, then the radius in the opposite direction is - 1.

Now this behaviour has a direct qualitative significance. Linear understanding is inherently dualistic (which represents the holistic meaning of 2). So the midpoint of the circle at 1 unit represents thereby half of the total line.

However to understand the measurement of the line from the centre of the circle in complementary positive and negative directions we require - literally - nondual insight (so that dual notions are rendered paradoxical). So the midpoint from this perspective is 0.

Properly understanding the Riemann Hypothesis requires both dual and nondual understanding. Indeed it points to the ultimate state where both dual and nondual notions (or alternatively quantitative and qualitative notions) of number are identical. And this state is utterly mysterious and thereby cannot be grasped in a phenomenal manner.

Thus the mysterious order governing the nature of the prime numbers is already inherent in all number behaviour representing a non-phenomenal reality (where qualitative cannot be distinguished from quantitative notions).

The attempt to prove the Riemann Hypothesis in a merely (reduced) quantitative manner is thereby utterly futile.

The significance of the Riemann Hypothesis is ultimately of a truly breath taking order in that - properly understood - phenomenal reality as we know it would not be even possible if it did not hold.

In other words underlying all of visible reality is reality is a secret code ensuring a perfect harmony of the prime numbers with respect to their specific (quantitative) and holistic (qualitative) interaction.

It is this harmony that enables all subsequent phenomenal events (representing varying interactions of this original prime number number code) to unfold.

This once again suggests that ultimately the underlying nature of reality - insofar as it can be phenomenally investigated - is purely mathematical!

In other words at the deepest level, phenomena represent the interaction of a fundamental mathematical code that governs the subsequent behaviour of all natural events. However though we can only come to knowledge of this code through phenomena, its nature - and indeed in reverse fashion what we know as nature - is ultimately ineffable.

So at some stage, Physics will have to abandon the quest for the ultimate particles before arriving at a purely mathematical appreciation of reality that underlies all manifestations of such particles.

Indeed properly understood it has already arrived at this point. As I would see it, string theory represents an elaborate fiction that the ultimate physical particles are strings. Strictly speaking however these have no manifest physical reality but really operate as the vehicle of ever purer mathematical notions.

However my key point all along is that the very scope of Mathematics needs to be radically extended to explicitly include both its qualitative as well as quantitative aspect. In this light all particles ultimately emerge as the dynamic interaction of a prime number mathematical code that is designed to preserve the perfect harmony of both its quantitative and qualitative aspects.

Though this point is largely lost because of the reductionist nature of current scientific understanding, the truly great wonder of reality is how both the finite and infinite - though utterly distinct - yet successfully coexist with each other at all levels of understanding.

Thus before we can even for example engage in conventional mathematical activity, we must already presume this meaningful correspondence of finite and infinite. Indeed as I have frequently pointed out it underlines the very notion of mathematical proof!

Looked at another way, the Riemann Hypothesis is the necessary condition for such correspondence to exist. So it represents in fact a massive act of faith in the subsequent meaning of the whole mathematical enterprise.

As the Riemann Hypothesis thereby already underlies conventional mathematical proof (as the starting condition for its meaningful interpretation), The Riemann Hypothesis cannot therefore be proven (in conventional terms).

However far from this representing a defeat, true realisation of this fact has the capacity to open up appreciation of a greatly enlarged scope for Mathematics where every number, symbol, relationship has a unique qualitative - as well as quantitative - significance. And with this will come an enormous enrichment of the true nature of both Mathematics and Science.

## Thursday, December 15, 2011

### The Harmonic Series Again!

As I have repeatedly stated Conventional (Type 1) Mathematics is formally based on a linear i.e. 1-dimensional rational approach (in qualitative terms).

However the considerable problem that exists is that actual understanding of all mathematical processes entails an interaction of both rational and intuitive type understanding.

Thus the conventional approach simply reduces the intuitive aspect in rational terms. Alternatively it inevitably reduces - in any context - qualitative to quantitative type interpretation.

In qualitative terms, rational understanding always implies the positing of phenomena in a conscious manner.

Intuitive understanding - by contrast - implies the corresponding dynamic negation of such understanding in an unconscious holistic manner.

Thus a full account of the nature of linear understanding (that entails both rational and intuitive type understanding) requires recognition of the 1st dimension in both a positive and negative manner.

I have pointed already to the important fact that the very notion of a fraction implicitly requires transition from the positive (rational) to negative (intuitive) recognition with respect to the 1st dimension.

Thus in quantitative terms 4 is more fully represented as 4^1.

In qualitative terms this represents rational (conscious) understanding with respect to 1st dimension (positive).

However the related fraction 1/4 is initially represented as 4^(- 1)and in qualitative terms this relates to intuitive (unconscious) understanding through negation of the 1st dimension (negative).

So the very process of switching from whole to part recognition in experience implicitly requires switching from rational to intuitive type recognition.

However because Conventional Mathematics is formally defined in qualitative terms with respect to the positive 1st dimension, the significance of this qualitative change in understanding is overlooked with the result interpreted in a merely (reduced) quantitative manner i.e. 1/4 as (1/4)^1. So though a qualitative holistic shift of consciousness is necessary to enable the transformation from whole to part recognition, both whole and part are then interpreted in a merely quantitative manner (i.e. with respect to the positive 1st dimension).

Many indications of what I am saying here are provided by simple mathematical results.

Once again from where conventional rational understanding is of a linear nature, holistic intuitive recognition is qualitatively of a circular nature.

Prime numbers from a rational perspective are the most linear of all numbers and - literally 1-dimensional in nature (with no composite factors).

So 7^1 is truly linear in this regard with no factors.

4 by contrast (as composite) is inherently 2-dimensional in nature i.e. 2^2.

However when we raise a prime number to - 1, a remarkable transformation takes place whereby it now exhibits highly circular characteristics.

So 7^(- 1) = 1/7 = .142857..

Here the 6 digits 142857 have some notable circular properties. For example they recur indefinitely in the same manner. Also when we multiply by 2, 3, 4, 5 and 6 the same digits appear (that maintain the same cyclical order).

So 1/7 is perhaps the best known of the cyclic primes (that exhibit such circular properties).

Then the sum of the natural numbers 1 + 2 + 3 + 4 + 5 +..... represents the archetypal linear series of numbers i.e. as numbers naturally marked off on a straight line.

So all these numbers are implicitly 1-dimensional i.e. raised to the power of 1.

However when we raise these numbers same numbers to - 1

i.e. 1^ (- 1), 2^(- 1), 3^(- 1), 4^(- 1), 5^(- 1) +.... we generate the harmonic series 1 + 1/2 + 1/3 + 1/4 + 1/5 + .....

Now remember my basic starting about regarding the nature of the primes is that they combine extreme characteristics with respect to linear and circular aspects!

So the individual prime numbers are the most independent and linear of all numbers (with no constituent factors).

However the general behaviour of the primes (in the frequency of their overall distribution) involves the other extreme of a holistic circular tendency.

Now the very manner in which both linear and circular aspect are involved in experience goes back to the way in which whole and part interact. So to switch from the whole say 4 (4^1) to part 1/4 the dimensional number switches to - 1. So 1/4 = 4^(- 1).

Thus the decisive switch from whole to part requires that conscious (linear) understanding that is defined with respect to the positive 1st dimension be dynamically negated in an unconscious (circular) intuitive manner.

So similar dynamics are involved with respect to the harmonic series (by comparison with the natural number series).

And remarkably the harmonic series gives the simplest answer to the general nature of prime number distribution. In other words the measure of the average spread as between successive prime numbers (which becomes progressively larger as the natural numbers increase) is given by the sum of the harmonic series!

Indeed the change in the average gap between these primes as n i.e n^(+ 1) increases by 1 is given as 1/n i.e. n^(- 1).

So here in the general behaviour of the prime numbers we have an intimate relationship as between the positing and negating respectively (with respect to the 1st dimension) of the number n.

So moving from the specific independent linear notion of an individual prime number to the holistic interdependent circular notion of prime number distribution directly involves the positing (in conscious rational manner) and the corresponding negation (in an unconscious intuitive manner) of linear (1-dimensional) understanding.

So the proper understanding in conventional (Type 1) mathematical terms of the quantitative nature of prime number behaviour is inseparable from corresponding holistic (Type 2) qualitative mathematical interpretation (of such behaviour).

And such understanding requires that we can combine both linear and circular type notions equally in both a quantitative and qualitative manner.

This circular nature of the harmonic series can be demonstrated in yet another striking manner.

Now the harmonic series represents one important example of the Zeta function

i.e. 1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) + .... where s = 1.

As is well known the for all (positive) even integer values of s the resulting sum of the series can be given in terms of a expression involving pi. And as pi serves as the direct relation of circular and linear quantitative notions in the relationship of the (circular) circumference to its (line) diameter, likewise pi serves as the archetypal relation of corresponding circular and linear notions (understood in a qualitative manner).

For example when s = 2, the sum of the series = {(pi)^2}/6.

The harmonic series can be expressed in terms of the combination of the sum of values corresponding with even integer values for the Zeta Function.

So 1 + 1/2 + 1/3 + 1/4 + ........

= 2{ζ(2)/2 + ζ(4)/4 + ζ(6)/6 + ζ(8)/8 + ......}

And this infinite series can therefore be expressed as the continual sum of terms that consist of powers of pi (that are multiplied by a rational fraction)!

It is again notable that the harmonic series is intimately associated with musical sound. So just as the natural numbers can again be seen as the supreme expression of linear quantitative understanding, the harmonic series can be seen in a sense as a supreme expression of qualitative type appreciation.

And ultimately this is related to the fact that in switching from the positive 1st dimension to its negative one likewise switches from (conscious) rational to (unconscious) intuition, which is the very means by which we switch from quantitative to qualitative type appreciation.

However the considerable problem that exists is that actual understanding of all mathematical processes entails an interaction of both rational and intuitive type understanding.

Thus the conventional approach simply reduces the intuitive aspect in rational terms. Alternatively it inevitably reduces - in any context - qualitative to quantitative type interpretation.

In qualitative terms, rational understanding always implies the positing of phenomena in a conscious manner.

Intuitive understanding - by contrast - implies the corresponding dynamic negation of such understanding in an unconscious holistic manner.

Thus a full account of the nature of linear understanding (that entails both rational and intuitive type understanding) requires recognition of the 1st dimension in both a positive and negative manner.

I have pointed already to the important fact that the very notion of a fraction implicitly requires transition from the positive (rational) to negative (intuitive) recognition with respect to the 1st dimension.

Thus in quantitative terms 4 is more fully represented as 4^1.

In qualitative terms this represents rational (conscious) understanding with respect to 1st dimension (positive).

However the related fraction 1/4 is initially represented as 4^(- 1)and in qualitative terms this relates to intuitive (unconscious) understanding through negation of the 1st dimension (negative).

So the very process of switching from whole to part recognition in experience implicitly requires switching from rational to intuitive type recognition.

However because Conventional Mathematics is formally defined in qualitative terms with respect to the positive 1st dimension, the significance of this qualitative change in understanding is overlooked with the result interpreted in a merely (reduced) quantitative manner i.e. 1/4 as (1/4)^1. So though a qualitative holistic shift of consciousness is necessary to enable the transformation from whole to part recognition, both whole and part are then interpreted in a merely quantitative manner (i.e. with respect to the positive 1st dimension).

Many indications of what I am saying here are provided by simple mathematical results.

Once again from where conventional rational understanding is of a linear nature, holistic intuitive recognition is qualitatively of a circular nature.

Prime numbers from a rational perspective are the most linear of all numbers and - literally 1-dimensional in nature (with no composite factors).

So 7^1 is truly linear in this regard with no factors.

4 by contrast (as composite) is inherently 2-dimensional in nature i.e. 2^2.

However when we raise a prime number to - 1, a remarkable transformation takes place whereby it now exhibits highly circular characteristics.

So 7^(- 1) = 1/7 = .142857..

Here the 6 digits 142857 have some notable circular properties. For example they recur indefinitely in the same manner. Also when we multiply by 2, 3, 4, 5 and 6 the same digits appear (that maintain the same cyclical order).

So 1/7 is perhaps the best known of the cyclic primes (that exhibit such circular properties).

Then the sum of the natural numbers 1 + 2 + 3 + 4 + 5 +..... represents the archetypal linear series of numbers i.e. as numbers naturally marked off on a straight line.

So all these numbers are implicitly 1-dimensional i.e. raised to the power of 1.

However when we raise these numbers same numbers to - 1

i.e. 1^ (- 1), 2^(- 1), 3^(- 1), 4^(- 1), 5^(- 1) +.... we generate the harmonic series 1 + 1/2 + 1/3 + 1/4 + 1/5 + .....

Now remember my basic starting about regarding the nature of the primes is that they combine extreme characteristics with respect to linear and circular aspects!

So the individual prime numbers are the most independent and linear of all numbers (with no constituent factors).

However the general behaviour of the primes (in the frequency of their overall distribution) involves the other extreme of a holistic circular tendency.

Now the very manner in which both linear and circular aspect are involved in experience goes back to the way in which whole and part interact. So to switch from the whole say 4 (4^1) to part 1/4 the dimensional number switches to - 1. So 1/4 = 4^(- 1).

Thus the decisive switch from whole to part requires that conscious (linear) understanding that is defined with respect to the positive 1st dimension be dynamically negated in an unconscious (circular) intuitive manner.

So similar dynamics are involved with respect to the harmonic series (by comparison with the natural number series).

And remarkably the harmonic series gives the simplest answer to the general nature of prime number distribution. In other words the measure of the average spread as between successive prime numbers (which becomes progressively larger as the natural numbers increase) is given by the sum of the harmonic series!

Indeed the change in the average gap between these primes as n i.e n^(+ 1) increases by 1 is given as 1/n i.e. n^(- 1).

So here in the general behaviour of the prime numbers we have an intimate relationship as between the positing and negating respectively (with respect to the 1st dimension) of the number n.

So moving from the specific independent linear notion of an individual prime number to the holistic interdependent circular notion of prime number distribution directly involves the positing (in conscious rational manner) and the corresponding negation (in an unconscious intuitive manner) of linear (1-dimensional) understanding.

So the proper understanding in conventional (Type 1) mathematical terms of the quantitative nature of prime number behaviour is inseparable from corresponding holistic (Type 2) qualitative mathematical interpretation (of such behaviour).

And such understanding requires that we can combine both linear and circular type notions equally in both a quantitative and qualitative manner.

This circular nature of the harmonic series can be demonstrated in yet another striking manner.

Now the harmonic series represents one important example of the Zeta function

i.e. 1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) + .... where s = 1.

As is well known the for all (positive) even integer values of s the resulting sum of the series can be given in terms of a expression involving pi. And as pi serves as the direct relation of circular and linear quantitative notions in the relationship of the (circular) circumference to its (line) diameter, likewise pi serves as the archetypal relation of corresponding circular and linear notions (understood in a qualitative manner).

For example when s = 2, the sum of the series = {(pi)^2}/6.

The harmonic series can be expressed in terms of the combination of the sum of values corresponding with even integer values for the Zeta Function.

So 1 + 1/2 + 1/3 + 1/4 + ........

= 2{ζ(2)/2 + ζ(4)/4 + ζ(6)/6 + ζ(8)/8 + ......}

And this infinite series can therefore be expressed as the continual sum of terms that consist of powers of pi (that are multiplied by a rational fraction)!

It is again notable that the harmonic series is intimately associated with musical sound. So just as the natural numbers can again be seen as the supreme expression of linear quantitative understanding, the harmonic series can be seen in a sense as a supreme expression of qualitative type appreciation.

And ultimately this is related to the fact that in switching from the positive 1st dimension to its negative one likewise switches from (conscious) rational to (unconscious) intuition, which is the very means by which we switch from quantitative to qualitative type appreciation.

## Wednesday, December 14, 2011

### A Strikingly Simple Prime Number Relationship

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.

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.

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