Monday, July 16, 2018

Differing Orders of Primes and Natural Numbers

I have raised before the notion of different primes with respect to the natural number system.

The well-known set of primes 2, 3, 5, 7, 11, … in this context can be referred to as Order 1 primes, with the corresponding natural numbers 1, 2, 3, 4, 5, …, in corresponding fashion referred to as Order 1 natural numbers.

However we can now order the cardinal primes 2, 3, 5, 7, 11, …, with respect to ordinal natural number rankings 1, 2, 3, 4, 5, …

If we now choose to list only those primes with a corresponding ordinal prime ranking we now obtain a new set of cardinal primes i.e. 3, 5, 11, 17, 31, 41, …

I then refer to this new set of cardinal primes as Order 2 primes with the corresponding set of natural numbers (including a) based on the use of these primes as factors as

1, 3, 5, 9, 11, 15, 17, 25, 27, 31, …

And I refer to this new set of natural numbers as Order 2 natural numbers.

We can now again give the Ordinal 2 cardinal primes an (Order 1) natural number ranking in an ordinal fashion i.e. 1, 2, 3, 4, 5, …

Then if we once again only choose those remaining primes that correspond with prime rankings we derive yet a new set of cardinal primes i.e. 5, 11, 31, 59, 127, …

I then refer to this latest set of primes as Order 3 primes.

And once more we can derive the corresponding Order 3 set of natural numbers (including 1) based on the use of these primes as factors i.e. 1, 5, 11, 25, 31, 55, 59, 125, 127, …

And we can continue on indefinitely in this fashion deriving an ever sparser set of primes and corresponding natural numbers (based on the use of these primes as factors).

For example the Order 4 primes are 11, 31, 127, … and the corresponding Order 4 natural numbers are 1, 11, 31, 121, 127, …

So we keep interchanging in this manner as between the quantitative notion of cardinal primes and the corresponding qualitative notion of ordinal natural number rankings.

Indeed one fascinating insight that derives from this approach is that the full set of natural numbers, can thereby be expressed as Order 0 primes!

So in this way the true independence as between the primes and natural numbers with respect to both analytic (quantitative) and holistic (qualitative) meanings is made readily apparent.


And what is very interesting in this regard is that we can derive an unlimited number of corresponding L-functions based on matching the Order k primes with the corresponding Order k natural numbers.

So for example, we can thereby match the Order 2 primes with the corresponding Order 2 natural numbers.

Therefore in general terms, with convergent answers where s > 1.

1 + 1/3s  + 1/5s + 1/9s  + 1/11s + …        = 1/(1 – 1/3s) * 1/(1 – 1/5s) * 1/(1 – 1/11s) * …

So for example, where s = 4,

1 + 1/34  + 1/54 + 1/94  + 1/114 + …        = 1/(1 – 1/34) * 1/(1 – 1/54) * 1/(1 – 1/114) * …

The LHS (for listed terms) = 1.01416…, whereas the RHS = 1.01419… 

So already the two results are very similar.


This latest procedure - entailing Order k natural numbers with corresponding Order k primes - is different from previous procedures where we eliminated a finite number of terms with respect to the RHS (product over primes) with then made consequent adjustments for all terms where the primes acted as constituent factors in the LHS (sum over the integers) expression.

In this case we are with each successive Order, eliminating an infinite series of individual prime terms with respect to the RHS with then consequent adjustments to the LHS natural numbers expression.

And the product over primes expression is no longer adjusted for a finite number of rational terms - but rather an infinite number - we can no longer preserve results of the form kπs (where k again is a rational number and s a positive even integer).

However with respect to new Order k expressions we can again adjust for a finite number of individual terms (with respect to the RHS prime expression).

So, as we have seen the Order 2 function is,

1 + 1/3s  + 1/5s + 1/9s  + 1/11s + …        = 1/(1 – 1/3s) * 1/(1 – 1/5s) * 1/(1 – 1/11s) * …

Therefore if we now eliminate the individual term related to 3 as prime in the RHS, then we must correspondingly eliminate all terms which include 3 as factor in the corresponding LHS expression.

So  1 + 1/5s + 1/11s + 1/17s + 1/25s + …  = 1/(1 – 1/5s) * 1/(1 – 1/11s) * 1/(1 – 1/17s) * …              

Thursday, July 12, 2018

Two-way Complementarity of L1 and L2 Functions

We have in the last few entries emphasised the dynamic complementary nature of the Riemann zeta function - which by extension applies to all L-functions – in the manner in which both the analytic (quantitative) and holistic (qualitative) nature of number interacts in two-way fashion with each other.

This also implies a corresponding dynamic complementarity as between both L1 and L2 functions.

As we have seen whereas the collective whole function (with respect to both RHS product over primes and LHS sum over the integers expressions) is expressed as an L1 function, then each individual term is then expressed as a corresponding L2 function.

And just as we have seen the collective whole L1 function can be given two alternative expressions, likewise each individual L2 function can likewise be given two alternative expressions.

So again with respect to our prototype L function i.e. the Riemann zeta function

  ∞                
 ∑ 1/ns   = ∏1/(1 – 1/ps)
n = 1           p = 2

So here we have the general collective L1 function (with corresponding LHS sum over the integers and RHS product over the primes expressions).

Thus again for example when s = 2,

1/12 + 1/22 + 1/32 + 1/42 + …      = 1/(1 – 1/22) * 1/(1 – 1/32) * 1/(1 – 1/52) * …

i.e. 1 + 1/4 +  1/9 + 1/16 + …     = 4/3 * 9/8 * 25/24 * …     = π2/6

However each individual term (with respect to both expressions) can be given a corresponding L2 definition (as an infinite type series) directly related to the specific natural number and prime in question.

So for example the first term i.e. 4/3, in the RHS expression is related to the prime 2 as
1/(1 – 1/22).

This can be then expressed in L2 terms as

1 + {1/22}1 + {1/22}2 + {1/22}3  + …

So notice the complementarity with the L1 function!

In the L1 the natural numbers represent the base aspect, whereas in the L2, the natural numbers represent the dimensional aspect of number.

Also in the L1, 2 - as the first – prime appears as a fixed dimensional number, whereas in the L2, appears as the fixed denominator of the base aspect of number.

Now equally, each individual term - apart from the first default term of 1 - of the sum over integers L1 expression, can be expressed as an L2 function.

So 1/4 = [1 + {1/22 + 1}1 + {1/22 + 1}2 + {1/22 + 1 }3  + …] – 1

Thus this is based on the second natural number 2.

And all other terms are then based on the third, fourth, fifth … natural numbers respectively.

We equally can express each individual term of the L1 function (with respect to both expressions) with an alternative L2 function.

So for example again the first individual term 4/3 - related to 2 - as the first individual term of the product over primes L1 function can be expressed as

1/{22C0} + 1/{(1/22 + 1)C1} + 1/{(1/22 + 2)C2} + 1/{(1/22 + 3)C3} + …

i.e. 1/4C0 + 1/5C1 + 1/6C2 + 1/7C3 + …

= 1 + 1/5 + 1/15 + 1/35 + …  = 4/3

And the 2nd term (above) with respect to the sum over the integers L1 function, can be expressed as an L2 function as follows

[1/{(22 + 1)C0} + 1/{(22 + 2)C1} + 1/{(22 + 3)C2} + 1/{(22 + 4)C3} + …] – 1

i.e. [1/5C0 + 1/6C1 + 1/7C2 + 1/8C3 + …] – 1 

= [1 + 1/6 + 1/21 + 1/56 + …] – 1   = 1/4

So the two-way complementarity that we have already seen as between the analytic (quantitative) and holistic (qualitative) aspects of number is replicated by a corresponding two-way relativity as between L1 and L2 functions.

So when each individual term, as in our example above, of the product over primes expression (which can be rendered as an L2 function) is given an analytic (quantitative) interpretation then the combined collective product of terms (which can be rendered as an L1 function) then is given a complementary holistic (qualitative) interpretation.

Then in reverse manner when each individual term, of the product over primes expression is given a holistic (qualitative) interpretation then the combined collective product of terms is then given a complementary analytic (quantitative) interpretation.

And this equally applies with respect to the sum over the integers expression.
Here, when each individual term, (which can be rendered as an L2 function) is given an holistic (qualitative) interpretation - where prime factors are viewed in a shared manner - the combined collective product of terms (which can be rendered as an L1 function) is then given a complementary analytic (quantitative) interpretation.

Finally, when each individual term, with respect to the sum over the integers expression, is given an analytic (quantitative) interpretation - where prime factors are now viewed in a relatively independent manner - the combined collective product of terms (which can be rendered as an L1 function) is then given a complementary holistic (qualitative) interpretation. 

Wednesday, July 11, 2018

Holistic Nature of Pi Relationship

As is well-known when s is a positive even integer, the value of ζ(s) can be represented in the form kπs (where k is a rational fraction).

Thus again in the best known case where s = 2

ζ(2) = 1/12 + 1/22 + 1/32 + 1/42 + …  = 4/3 * 9/8 * 25/24 * 49/48 * …   = π2/6

And as we have seen the general form of this relationship i.e. kπs will continue to hold when we eliminate on the RHS (product over primes expression) an individual term or series of terms (based on a particular prime or series of primes) and then in corresponding fashion eliminate with respect to the LHS (sum over integers expression) the collection of terms where this prime or series of primes operate(s) as shared common factor(s).

Now this is already known since the time of Euler from an analytic (quantitative) perspective.

However what is fascinating - and indeed very important - is the fact that a convincing holistic (qualitative) mathematical rationale can likewise be given for the relationship.

And from the comprehensive dynamic interactive perspective of understanding, both analytic and holistic interpretations are necessarily involved.

When we look at π from the conventional quantitative perspective its value is explained as the (perfect) relationship of the circular circumference to its line diameter.  

However what is not all properly realised is that - as with all mathematical symbols - a corresponding holistic (qualitative) interpretation can be given.

So from this perspective π represents the (perfect) relationship of circular to linear type understanding.

And this is precisely what I have been striving to explain in the last few entries where the very interpretation of the related product over primes and sum over the integers expressions (which necessarily must hold with respect to the Riemann zeta function and all subsequent L-function) in fact entails in dynamic interactive terms the two way relationship of circular to linear type understanding.

So the standard analytic (quantitative) interpretation of the primes and natural numbers is to literally treat all members as lying on the real line.

However the largely unrecognised holistic (qualitative) interpretation of the primes and natural numbers entails treating them as inter-related points on the unit circle (in the complex plane).

Therefore for example with respect the first prime i.e. 2 is viewed in analytic (quantitative) terms as a point on the real number line (measuring from the origin two units).

So again this is the cardinal notion of number as composed of independent homogeneous units that corresponds to standard rational interpretation.

However 2 is viewed in corresponding holistic (qualitative) terms as composed in ordinal terms of unique 1st and 2nd members that are freely interchangeable with each other.
Whereas the direct qualitative recognition of this ordinal relationship requires intuitive insight (as a direct representation of psycho-spiritual energy), indirectly it is then represented in quantitative fashion as two equidistant points on the unit circle (in the complex plane) i.e. + 1 and – 1 respectively.

When we for example one recognises that that the two turns at a crossroads can potentially be left or right (depending on the direction from which the crossroads is approached) then one is I fact implicitly giving recognition to the holistic (qualitative) notion of 2.

So if from one direction of approach the left turn is designated as + 1 then in this context the right is thereby – 1 (not a left turn). If however from the opposite direction of approach the right turn is now designated as +1, then in this context the left is – 1 (not a right turn).

Thus potentially before a direction of approach is given, left and right turns have a purely circular (paradoxical) meaning, which directly equates with holistic (qualitative) understanding.

However once a fixed direction of approach is given then left and right acquire in actual terms as separate independent (as two unambiguously distinct turns + 1 and + 1).  

However potentially before a direction of approach is given left and right remain interdependent as, relatively,  + 1 and – 1 respectively.

So when reason and intuition are properly integrated in understanding both analytic (quantitative) and holistic (qualitative) aspects of all mathematical relationships must be explicitly incorporated.  

And again we can see the enormous reductionism which defines conventional mathematical interpretation i.e. where the qualitative aspect though of a radically distinctive intuitive nature is reduced in a merely rational manner.

Alternatively we can say that both linear and circular aspects must be perfectly combined (where circular in this context represents the indirect quantitative means of representing understanding that is directly of an intuitive i.e. qualitative nature)

So to properly understand the Riemann Zeta function and the fact that ζ(s) = kπs , where s is an even positive integer, it is not sufficient in the manner of Euler to provide a merely analytic (quantitative) interpretation of this relationship; equally in fact it is important to provide the corresponding holistic (qualitative) interpretation of the relationship, which I have been outlining in the last few entries.

For in fact, properly understood in dynamic interactive terms both interpretations are in truth inextricably inter-related.  

Finally with respect to this contribution I wish to provide a simple holistic explanation as to the perplexing fact, of which Euler was well aware (in quantitative terms) that where s is an positive odd integer that no relationship of the form ζ(s) = kπs  exists.

The key to the holistic appreciation of this apparent conundrum is the very notion of complementarity that always implies for any positive designation an exactly matching negative designation.

We can see this very simply with respect to the root structure of number.

As we have seen the n roots of 1 can be represented as n equidistant points on the unit circle (in the complex plane).

Now when n is even any root (with respect to one quadrant of the circle) can be directly connected through a diameter line drawn through to the centre of the circle to another root (in the opposite quadrant).

We can see this for example very simply with respect to the 4 roots of 1.

The two real roots + 1 and – 1 can be connected through the (horizontal) diameter line, whereas the two imaginary roots + i and – i can likewise be connected through the (vertical) diameter line.

And where this is the case - as with all even numbered roots - a direct complementary relationship thereby exists as between opposite roots.

However where we have an odd number of roots this is not the case.

Again we can see this simply in the case of the 3 roots of 1, where there is no way of connecting the different roots through a straight line diameter.

So in the case of odd numbered roots, a directly complementary relationship does not exist as between opposite roots. However when n is very large a relationship that progressively better approximates to perfect complementarity exists.

Tuesday, July 10, 2018

Complementary Connections Explained (1)

In yesterday’s blog entry, I illustrated the relationship as between each individual term (in the product over primes expression) with the collective nature of terms (in the corresponding sum over natural numbers expression).

And the crucial point here is that from the appropriate dynamic perspective that a complementary relationship exists so that what is interpreted in an quantitative manner with respect to each individual term (related to a specific prime), is then interpreted in a qualitative manner with respect to the collection of terms (where this specific prime operates as a shared common factor).

However as always with such dynamic complementary relationships reference frames can be switched, so that equally each individual term (with respect to the product over primes expression) can be viewed in a qualitative manner. Then when this is the case the corresponding interpretation with respect to the collection of natural numbers where the prime operates as a factor is in an quantitative manner.

So let me now illustrate this point at greater length with respect to the example yesterday of ζ(2), where we focussed on the first individual term in the RHS product over primes expression i.e. 4/3, which is based on the prime number “2”.

Now as we have seen in other entries, 2 as a prime can be given both analytic (quantitative) and holistic (qualitative) interpretations, which in the dynamics of understanding keep interchanging with each other.

Thus when we define 2 in cardinal terms as composed of independent homogeneous units (that are thereby devoid of qualitative distinction) i.e. 2 = 1 + 1, we thereby interpret this prime in an analytic (quantitative) manner.

However when we equally recognise that 2 is necessarily composed of both a 1st and 2nd unit, which are interchangeable with each other i.e. where each unit now is given a distinct qualitative identity in ordinal terms, then we interpret this prime in a corresponding holistic (qualitative) manner.

Once again, from the dynamic interactive perspective, both of these interpretations are relative, so that the analytic (quantitative) aspect of “2” cannot have any strict meaning in the absence of the corresponding holistic (qualitative) aspect; equally the holistic (qualitative) aspect of “2” cannot have any strict meaning in the absence of the corresponding analytic (quantitative) aspect. 

And of course this dynamic interactive nature equally applies to all other primes, where both analytic (quantitative) and holistic (qualitative) aspects are necessarily related to each other in a relative fashion.

So again referring to the RHS product over primes expression, we can start by defining each prime with respect to its holistic (qualitative) aspect where component units are understood as interdependent (and fully interchangeable) with each other. 

So from this perspective the prime number “2” (to which the first individual term relates) can be given a holistic (qualitative) interpretation.

Then with respect to the corresponding sum over the integers expression, 2 as a factor of all even terms is given a complementary analytic (quantitative) interpretation.

Clearly therefore, though in the earlier case “2” is viewed in holistic terms as a common factor (of all even numbers) it equally enjoys a relatively independent identity (where it can be viewed as separate from other factors). And it is this latter quantitative interpretation of a factor that is now emphasised in this context.

So overall each individual term (with respect to the product over primes expression) related to a specific prime, can be given both an analytic (quantitative) and holistic (qualitative) interpretation.

Likewise the collection of terms (with respect to the sum over integers expression) which includes a specific prime as factor, can be given both analytic (quantitative) and holistic (qualitative) interpretations.

But once again the key requirement in correct dynamic interactive terms is that the relationship between both expressions is of a complementary nature (with analytic complementing holistic and holistic complementing analytic interpretations respectively).

And in this latter case where each individual term (with respect to the product over primes expression) is given a holistic (qualitative) interpretation, then the collective product of these terms is now given a complementary analytic (quantitative) interpretation and vice versa so that when the terms (with respect to the sum over integers expression) that share a specific prime as factor is given a holistic (qualitative) interpretation, then the additive sum of terms is - relatively - of an analytic (quantitative) nature.

So in dynamic interactive terms, as we move from collective (whole) to an individual (part) identity or in reverse from an individual (part) to a collective (whole) identity, the very nature of number likewise keeps switching as between both its analytic (quantitative) and holistic (qualitative) aspects, in a relative complementary manner.

Thus the notion that number somehow can be properly interpreted in an absolute - merely quantitative - manner is utterly misleading and only valid as an important special case in a restricted limited context.
And it is this misleading notion of number - which ultimately is not fit for purpose - that has dominated the development of Mathematics over several millennia.   

Monday, July 9, 2018

Illustrations of Complementary Analytic and Holistic Connections

Once again the Riemann zeta function - which serves as the prototype for all L-functions - can be expressed both as a sum over all the positive integers and a product over all the primes.

So when s is an integer (> 1), both expressions converge with
                       
 ∑ 1/ns   = ∏1/(1 – 1/ps)
n = 1             p = 2

Thus in the well-known case, where s = 2,

1/12 + 1/22 + 1/32 + 1/42 + …      = 1/(1 – 1/22) * 1/(1 – 1/32) * 1/(1 – 1/52) * …

i.e. 1 + 1/4 +  1/9 + 1/16 + …     = 4/3 * 9/8 * 25/24 * …     = π2/6

Now each (individual) prime-based term on the RHS expression is linked to the corresponding (collective) natural number expression in a precise manner, where dynamic complementarity as between analytic (quantitative) and holistic (qualitative) is key in terms of appropriate interpretation.

So the first (individual) term in the RHS expression i.e. 4/3 is related in turn to first prime i.e 2.

Now when we give “2” its customary analytic interpretation in an independent quantitative manner with respect to the RHS expression, then in a dynamic complementary manner the “2” now acquires a corresponding holistic interpretation with respect to the LHS expression where it acts as a shared factor of all the even numbers.

So there are two distinctive meanings associated here with “2” that alternate as between a quantitative identity (as independent of other numbers) and a corresponding qualitative identity (as shared with other numbers) respectively.

The customary accepted meaning of “2” is in quantitative terms (as independent of other numbers). However, as we have seen when 2 operates as a common factor its meaning is thereby shared with all even numbers. Thus in this context the meaning of “2” is now of a qualitative nature (as related to all even numbers).

And when one reflects on the matter in the appropriate dynamic interactive manner, then it becomes quickly apparent that the interpretation of any number (such as 2) strictly must be of a merely relative nature, with a quantitative aspect (as relatively independent) and a qualitative aspect (as relatively interdependent) respectively.

Thus the standard treatment of integers as absolute quantities directly reflects the limited - and ultimately distorted - nature of conventional mathematical interpretation, where the distinctive qualitative aspect of number is reduced in a merely quantitative manner.


Now if we choose to eliminate the first (individual) term in the RHS expression (based on the prime 2), then we must correspondingly eliminate in a collective manner all the even numbered terms (where 2 is now a shared factor of these numbers).

Thus 1/12 + 1/32 + 1/52 + 1/72 + …  = 1/(1 – 1/32) * 1/(1 – 1/52) * 1/(1 – 1/72) * …

And since we have removed 4/3 from the previous RHS expression, the value of this new expression = (π2/6)/(4/3) = (π2 * 3)/24  = π2/8.

This new L-function - representing a sum over the odd-numbered positive integers - is referred as the Dirichlet Lambda Function γ(s), with

γ(s) = (1 – 2s) ζ(s)

So as we have seen when s = 2,

γ(2) = (1 – 1/4) ζ(2) = (3/4) * (π2/6)  = π2/8.


So the Dirichlet Lambda Function represents the special case where elimination of the first prime “2” is considered with respect to both the RHS expression (as an independent individual term) and the corresponding LHS expression (as a collective shared term of all even numbers).

However the same procedure can be applied to any of the primes.

So for example if we now decide instead of eliminating the first individual term in the RHS expression (based on 2) but rather the second term (based on 3), then we must in corresponding fashion eliminate all terms in the LHS expression (where 3 is a shared factor).

Thus 1/12 + 1/22 + 1/42 + 1/52 + …  = 1/(1 – 1/22) * 1/(1 – 1/52) * 1/(1 – 1/72) * …

And as the eliminated term in the product over primes expression is 9/8, then the corresponding value of this new L-function = (π2/6)/(9/8) = (π2 * 8)/54 = 4π2/27.

So this now leads to a new Lambda Function” which I will denote as γ3(s).

Thus γ3(s) = (1 – 2s) ζ(s).

And when again s = 2,

γ3(2) = (1 – 32) ζ(2) = 8/9 * π2/6 = 4π2/27

 

So we can derive a potentially unlimited number of new L-functions by successively eliminating in turn the term individually associated with any particular prime in the RHS expression and then collectively eliminating all terms in the LHS expression where this prime operates as a shared factor.

And in all cases the value of these functions (with s a positive even integer) will be in the form kπs (where k is a rational number).


We can also eliminate any desired combination of prime-based terms individually with respect to the RHS expression and then in corresponding fashion collectively eliminate all terms in the LHS expression (where any of these primes acts as a common factor).

For example if we now eliminate the two terms in the RHS expression, based on the primes 2 and 3 respectively, we then must correspondingly eliminate all terms in the LHS expression where 2 or 3 (or both) are factors.

Thus in general terms,

1/1s + 1/5s + 1/7s + 1/11s + …  = 1/(1 – 1/5s) * 1/(1 – 1/7s) * 1/(1 – 1/11s) * …

In fact the first non prime term in the LHS expression (where neither 2 nor 3 are factors) is 1/25s.

So when again s = 2, we have,

1/12 + 1/52 + 1/72 + 1/112 + …  = 1/(1 – 1/52) * 1/(1 – 1/72) * 1/(1 – 1/112) * …

And the value of this new L-function = π2/6 * 3/4 * 8/9  = π2/9.

And if we refer to this new Lambda Function” as γ23(s), then

γ23(s) = (1 – 2s)(1 – 3s)ζ(s).

And once again we can potentially have an unlimited number of combinations of primes which, when individually eliminated from the RHS product over primes expression then in corresponding fashion lead to the collective elimination in the LHS sum over integers expression, of all numbers based on any of these primes (or combination of primes) as factors.

Wednesday, June 27, 2018

Alt L2 Function

Just as the L1 function can be given two expressions i.e. sum over the positive integers and product over the primes respectively, equally the L2 function can be given two expressions.

In previous blog entries I have referred to this latter expression (which has its roots in the L1 function) as the Alt L2 function.

Now with respect to the Riemann zeta function, a special case occurs where s = 1, yielding the harmonic series,

1/1 + 1/2  + 1/3 + 1/4 + …

This can then be expressed in terms of combinations as

1/1C0 + 1/2C1 + 1/3C2 + 1/4C3  + …

Now, with respect to the denominator, we can define a new series where the nth term represents the sum of the n terms of the previous series.

So we thereby obtain

1/1 + 1/(1 + 2) + 1/(1 + 2 + 3) + 1/(1 + 2 + 3 + 4) + …

= 1 + 1/3 + 1/6 + 1/10 + …

Now this can equally be expressed in terms of combinations,

i.e. 1/2C0 + 1/3C1 + 1/4C2 + 1/5C3  + …

So the get this new series, we only need to increase the starting number of each combination (in relation to the previous harmonic series) by 1.

Now if we look at this later combination expression (which starts with 1/2C0), we find that its value is equal to the corresponding L2 function i.e. 1 + x1 + x2  + x3  +…, where x = 1/2.

So 1 + 1/2 + 1/22  + 1/23 + …   = 2, and  

1 + 1/3 + 1/6 + 1/10 + …  = 2.


And this is universally the case for any integer k (k > 1, where x = 1/k) .

So when x = 1/k, then the L2 function i.e.

1/(1 – 1/k) = 1 + 1/k + 1/k2  + 1/k3 + …  has a corresponding Alt L2 expression as,

1/kC0 + 1/{(k + 1)C1}  + 1/{(k + 2)C2} + 1/{(k + 3)C3} + …

Therefore when for example k = 4,

1/(1 – 1/4) = 1 + 1/4 + 1/42  + 1/43 + …  = 4/3.

So this represents the appropriate L2 function in this case where x = 1/22 = 1/4.

And this can be given an Alt L2 expression as,

1/4C0 + 1/5C1 + 1/6C2 + 1/7C3 + …

= 1 + 1/5 + 1/15 + 1/35 + …  = 4/3

In fact a fascinating product type expression can also be given for this Alt L2 function.

So when as in this case the starting value of the combination is 4, we let the numerator and denominator of the 1st term = 4 * 3 * 2 * 1.

Thus the 1st term = (4 * 3 * 2 * 1)/(4 * 3 * 2 * 1) = 1.

Then to get the next term we simply add 1 to each value in the denominator while holding the numerator fixed.

So the 2nd term = (4 * 3 * 2 * 1)/(5 * 4 * 3 * 2) = 1/5.

And for each subsequent term, we keep repeating by adding 1 to each previous value in the denominator (while holding the numerator fixed).

So the 3rd term = (4 * 3 * 2 * 1)/(6 * 5 * 4 * 3)  = 1/15,

and the 4th term = (4 * 3 * 2 * 1)/(7 * 6 * 5 * 4)  = 1/35 and so on.

And the existence here of 4 product terms (in relation to numerator and denominator) coincides with the fact that the value of k (with respect to the L2 function) = 4.

Friday, June 22, 2018

Complementary L1 and L2 Functions Continued

In the last entry, I indicated how each individual term, in the product over primes expression of the L function, can be represented as a complementary L function (where the base and dimensional aspects of number are inverted).

I thereby refer to the former collective function as the L1 function and the latter individual function (representing a single term) as the L2 function.

So the (collective) L1 function is comprised of an infinite series of (individual) L2 functions. And once again these operate in a dynamic complementary manner with respect to each other.

So again in general terms (s > 1), with respect to the Riemann zeta function, the L1 function is customarily represented (in its sum over the positive integers form) as

1– s + 2– s + 3– s + 4– s  + …,

with the corresponding L2 function represented as

1 + (1/ps)+ (1/ps)+ (1/ps)+ (1/ps)+ …,

where p is defined over all the primes.  

However it is not only each term of the product over primes that can be represented as an L2 function, but likewise - with minor adjustment - each term of the sum over natural numbers expression.

The 1st term (in the sum over positive integers expression) as 1 stands alone, as it were, without a corresponding L2 expression.

However every other individual term can then be expressed in a general manner as an L2 function as follows.

[1/{1 1/(ks + 1)}] – 1 where k = 1, 2, 3, 4, …

= [{1 + {1/(ks + 1)}1 + {1/(ks + 1)}2 + {1/(ks + 1)}3 + {1/(ks + 1)}4 + …] 1 = 1/ks.

For example for ζ(2), where p = 2, the 2nd term is,

[{1 + {1/(22 + 1)}1 + {1/(22 + 1)}2 + {1/(22 + 1)}3 + {1/(22 + 1)}4 + …] 1,

= [1 + 1/5 + 1/52  + 1/53 + 1/54  + …] 1,

= 5/4 1  = 1/22  = 1/4.

So just as the 1st term 1, (i.e. + 1) is omitted in terms of a L2 expression, every other term includes the addition of + 1 , in its denominator expression.   

In this context it is fascinating to observe the case where both L1 and L2 functions, with respect to the Riemann zeta function in the analytically continued complex plane, diverge.

For the L1, with s = 1 (representing the dimensional aspect of number as exponent) we have the harmonic series

1/11 + 1/21 + 1/31 + 1/41 + … = ∞.

Then with the L2 when s = 1 (representing the corresponding base aspect of number) we have,

1/(1 1) = 1 + 11 + 12 + 13  + …  = ∞.

Thus from a dynamic interactive perspective, there is both an L1 and L2 explanation for the one pole in the Riemann zeta function i.e. s = 1 (where the functions are not defined).

So for the L1 function this pole occurs where s (representing the dimensional aspect of number) = 1.

For the L2 function, this pole occurs in complementary fashion where s (representing the corresponding base aspect of number) = 1.