More Connections

It struck me clearly the other day that each L2 function can be expressed as the quotient of two L1 functions.

For example in the Riemann zeta function (where s = 2) the first term in the product over primes expression is 4/3, which is related to the corresponding first prime 2, through the relationship 
1 – 1/2².

Now this can be expressed as the quotient of the functions,

1/1² + 1/2² + 1/3²  + 1/5² + … and 1/1²  + 1/3² + 1/5²  + 1/7² + … (where in the latter case all numbers which contain 2 as a shared factor are omitted).

So (1/1² + 1/2² + 1/3²  + 1/5² + …)/(1/1²  + 1/3² + 1/5²  + 1/7² + …)

= 1 + (1/2²)¹ + (1/2²)² + (1/2²)³ + …    = 4/3


And notice the complementarity here! Whereas the natural numbers represent the base aspect in the L1, they represent the dimensional aspect of number in the L2; and whereas 2 is constant as a dimensional number in the L1 it is constant (as reciprocal) in L2. So this properly signifies the dynamic interactive relationship connecting both functions.

By definition, any of the individual terms (in the product over primes expressions) or indeed combination of terms - which can be expressed as an L2 function or product of L2 functions - can be expressed as the quotient of two L1 functions.

For example the second term in the Riemann zeta function (where s = 3) is 27/26 relating to 1 – 1/3³.

This can be expressed as,

(1/1³ + 1/2³ + 1/3³ + 1/4³ + …)/(1/1³  + 1/2³ + 1/4³ + 1/5²  + …)

= 1 + (1/3³)¹ + (1/3³)² + (1/3³)³ + …    = 27/26

And if for example the first two terms 8/7 and 27/26 relating to the primes 2 and 3 respectively, were to be omitted, we would have,

(1/1³ + 1/2³ + 1/3³ + 1/5³ + …)/(1/1³  + 1/5³ + 1/7³ + 1/11³  + …)

= {1 + (1/2³)¹ + (1/2³)² + (1/2³)³ + …} * {1 + (1/3³)¹ + (1/3³)² + (1/3³)³ + …}

I mentioned in a previous blog entry that the generalised expression for the Riemann zeta function linking sum over the integers and product over the primes expressions is

1/1^s + (1 + k₁)/2^s + (1 + k₂)/3^s + (1 + k₁)/4^s + (1 + k₃)/5^s + {(1 + k₁)(1 + k₂)}/6^s + …
= 1/{1 – (1 + k₁)/(2^s + k₁)} * 1/{1 – (1 + k₂)/(3^s + k₂)} * 1/{1 – (1 + k₃)/(5^s + k₃)} * …,
 

Then when k₁ = k₂ = k₃,… = 1 that when s is an even number that a rational value results for the corresponding expressions.

So again when s = 2 and k₁ = k₂ = k₃,… = 1, then

1/1² + 2/2² + 2/3² + 2/4² + 2/5² + 4/6² + 2/7² + 2/8² + 2/9² + 4/10² + …

= 5/3 * 10/8 * 26/24 * 50/48 * …    = 5/2

Now if we eliminate any of the individual product over primes terms, then a resulting Dirichlet L-series will again result with a rational number value.

So if we eliminate the first term (in the product over primes expression) i.e. 5/3 (relating to 2) then we have,

1/1² + 2/3² + 2/5² + 2/7² + 2/9² + 2/11² + … = 10/8 * 26/24 * 50/48 * 122/120 * … = 3/2

And if alternatively we eliminate the second term (in the product over primes expression) i.e. 10/8 (relating to 3) we have

1/1² + 2/2² + 2/4² + 2/7² + 2/8² + 2/10² + … = 5/3 * 26/24 * 50/48 * 122/120 …   = 2

Intertwining L1 and L2 Functions

As is well known when s = 4,

ζ(4) = 1/1⁴ + 1/2⁴  + 1/3⁴ + 1/4⁴ + …   = π⁴/90

and when s = 2,

ζ(2) = 1/1² + 1/2²  + 1/3² + 1/4² + …    = π²/6, so that

(1/1² + 1/2²  + 1/3² + 1/4² + …)²  = π⁴/36  

Therefore (1/1² + 1/2²  + 1/3² + 1/4² + …)²/(1/1⁴ + 1/2⁴  + 1/3⁴ + 1/4⁴ + …)  = 90/36 = 5/2

However we have already seen that this can be written as a Dirichlet L-series i.e.

1/1² + 2/2² + 2/3² + 2/4² + 2/5² + 4/6² + 2/7² + 2/8² + 2/9² + 4/10² + … = 5/2

So ζ(2)²/ζ(4) = 1/1² + 2/2² + 2/3² + 2/4² + 2/5² + 4/6² + 2/7² + 2/8² + 2/9² + 4/10² + …

Thus we have expressed the square of one Dirichlet series (the Riemann zeta function where s = 2) divided by another Dirichlet series (the Riemann zeta function where s = 4) by yet another Dirichlet series (where s = 2).

And this relationship can equally be expressed as a product over primes expression.

So ζ(2)²/ζ(4)  = (4/3 * 9/8 * 25/24 * 49/48)²/(16/15 * 81/80 * 625/624 * 2301/2300)

= 5/3 * 10/8 * 26/24 * 50/48 * …

However in the terms that I employ, this collective relationship (for both expressions) relates solely to the L1 function.

So the question now arises as to how each individual term can be expressed as a corresponding (infinite) L2 function.

Thus with respect to the geometric series type expression, if we take the first term, i.e. 5/3, we can express this as the product of two terms i.e. 5/4 * 4/3.

And then each of these terms in turn can be expressed in the standard manner (related to the first prime 2).

So 5/4 = 1 + 1/(2² + 1)¹  + 1/(2² + 1)² + 1/(2² + 1)³ + …

And 4/3 = 1 + (1/2²)¹  + 1/(1/2²)² + 1/(1/2²)³ + …

So just as we have expressed the L1 function as the quotient of two other functions, now in complementary fashion, we have expressed the L2 as the product of two functions.

We could also express 5/3 directly as an L2 (geometric series) function in the following way,

5/3 = 1 + {2/(2² + 1)}¹  + {2/(2² + 1)}² + {2/(2² + 1)}³ + …

So again just as we can express the quotient of the two L1 functions in a direct manner equally we can express the product of the two L2 functions equally in a direct L2 manner.

Finally this equally applies to the Alt L2 function.

So once again 5/3 = 5/4 * 4/3

And each of these can be expressed as an Alt 2 function.

So   5/4 = 1 + 1/6 + 1/21 + 1/ 56 + …

And 4/3 = 1 + 1/5 + 1/15 + 1/35 + …

So 5/3 thereby represents the product of both these Alt L2 functions.

However again - as with the geometric series form - a direct Alt L2 expression can be found.

Here let 5/3 = (n – 1)/(n – 2)

So n = 3.5 which acts as the denominator of the 2^nd term.

As the 1^st term is always 1 then this implies that that the 2^nd term is 1/3.5 = 2/7.

Then to get the 3rd term multiply 7/2 by (3.5 + 1)/2 = 7/2 * 9/4 and get the reciprocal i.e. 8/63. Then for the next term multiply 63/8 by (3.5 + 2)/3 and get reciprocal = 16/231, and continue on in this fashion in each case adding an additional 1 to both numerator and denominator of the multiplying number.

So the first 4 terms of the series are    

1 + 2/7 + 8/63 + 16/231 + …   = 5/3.

And then with respect to the sum over the integers expression each individual term can again be expressed both in an L2 (geometric series) and Alt L2 fashion by a similar approach (where 1 is subtracted in each case).

So for example the 7th term, i.e. 2/7²  = 2/49.

Thus we obtain the L2 and Alt L2 functions for 51/49 (and then finally adjust by subtracting 1 in each case).

The L2 function is {1 + [2/(7² + 2)]¹ + [2/(7² + 2)]² + [2/(7² + 2)]³ + ...} – 1

The Alt L2 function is {1 + 2/53 + 8/2915 +16/55385 + ...} – 1. 

Another Interesting Generalisation (2)

In the last entry, I mentioned the new general formula, which can be used to extend relationships as between sum over integers and product over primes expressions with respect to the Riemann zeta function i.e.

1/1^s + (1 + k₁)/2^s + (1 + k₂)/3^s + (1 + k₁)/4^s + (1 + k₃)/5^s + {(1 + k₁)(1 + k₂)}/6^s + …
= 1/{1 – (1 + k₁)/(2^s + k₁)} * 1/{1 – (1 + k₂)/(3^s + k₂)} * 1/{1 – (1 + k₃)/(5^s + k₃)} * …,
 
where k₁, k₂, k₃, … are rational numbers which can be either positive or negative.

In that entry, I demonstrated the - surely - interesting fact that when s is 2, 4, 6, …

and k₁ = k₂ = k_3, …  = 1, that the value of the two expressions (sum over the integers and product over the primes respectively) is a rational number.

In this case there is a clear relationship as between the values of the standard zeta function for s and 2s respectively.

So - again when s is a positive even integer - the value of the zeta expression (for both sum over the integers and product over the primes expression) is t₁π^s (where t₁ is a rational number).

Likewise the value of the expression when the dimension is 2s is t₂π^2s (where t₂ is a rational number).

Let the ratio t₂/t₁  = t_3.

Then the value of the extended zeta function where k₁ = k₂ = k_3, …  = 1, is t₃/t₁ i.e. t₂/(t₁)².

So when s = 2, ζ(2) = π²/6 and when s = 4, ζ(4) = π⁴/90.

So t₁ = 6 and t₂  = 90.

Therefore t₂/(t₁)²  = 90/36 = 5/2

And this is the value of the extended expression where k₁ = k₂ = k_3, …  = 1.


Another interesting - if trivial - case arises when k₁ = k₂ = k_3, …  = – 1.

This leads to the elimination on both sides of all terms entailing k so that we are left with the identity 1 = 1.

This is useful to remind us that 1 precedes all subsequent relationships entailing prime factors. So 1 is not obtained from these relationships but rather serves as a necessary precondition for their use. 

Thus internally the very notion of a prime entails unit components which then have both a quantitative identity as (independent of each other) and a qualitative identity as (interdependent with other) respectively.

So far in the use of the extended general formula we have shown how to generate expressions where the numerator  > 1.

However by picking the value of k₁, k₂, k_3, …, appropriately we can likewise generate expressions where the denominator - rather than the numerator term -  increases.

For example when k₁ = – 1/2, k₂ = – 2/3, k₃ = – 4/5, …, we have 

1/1^s + (1 – 1/2)/2^s + (1 – 2/3)/3^s + (1 – 1/2)/4^s + (1 – 4/5)/5^s + {(1 – 1/2)(1 – 2/3)}/6^s + …

= 1/{1 – (1 – 1/2)/(2^s – 1/2)} * 1/{1 – (1 – 2/3)/(3^s – 2/3)} * 1/{1 – (1 – 4/5)/(5^s – 4/5)} * …,

i.e.

1/1^s + 1/(2.2^s) + 1/(3.3^s) + 1/(2.4^s) + 1/(5.5^s) + 1/(6.6^s) + …

= 1/1 – {(1/2)/(2^s – 1/2)} * 1/1 – {(1/3)/(3^s – 2/3)} * 1/1 – {(1/5)/(5^s – 4/5)} * …,

So for example, when s = 2

1/1² + 1/(2.2²) + 1/(3.3²) + 1/(2.4²) + 1/(5.5²) + 1/(6.6²) + …

= 1/{1 – [(1/2)/(2² – 1/2)]} * 1/{1 – [(1/3)/(3² – 2/3)]} * 1/{1 – [(1/5)/(5² – 4/5]} * …,

= 1/(1 – 1/7) * 1/(1 – 1/25) * 1/(1 – 1/121) * …= 7/6 * 25/24 * 121/120 * …

And likewise k₁, k₂, k_3, …, can be chosen so that both numerator and denominator are multiples of values that occur in the standard case.

Another Interesting Generalisation (1)

It just struck recently that the sum over the integers and product over the primes expressions, with respect to the Riemann zeta function, represent a special case of a more comprehensive relationship, which can be expressed in general terms as follows:

1/1^s + (1 + k₁)/2^s + (1 + k₂)/3^s + (1 + k₁)/4^s + (1 + k₃)/5^s + {(1 + k₁)(1 + k₂)}/6^s + …

= 1/{1 – (1 + k₁)/(2^s + k₁)} * 1/{1 – (1 + k₂)/(3^s + k₂)} * 1/{1 – (1 + k₃)/(5^s + k₃)} * … 

where k₁, k₂, k₃, … are rational numbers which can be either positive or negative.


So the standard equation represents the situation where the values for  k₁, k₂, k₃, … = 0_. 

It can be seen here it is the distinct prime factors of a number that determines the nature of the sum over the integers expression.

So for example, 6 is the 1st composite natural number with two distinct prime factors. Therefore as we can see from the new formula we must here thereby multiply 1 + k₁ (associated with 2) by 1 + k₂ (associated with 3) respectively.

One very simple case with respect to this general formula is for k₁ = 1 and k₂, k₃, k₄, … = 0

Then,

1/1^s + 2/2^s + 1/3^s + 2/4^s + 1/5^s + …  =  1/{1 – 2/(2^s + 1)} *  1/(1 – 1/3^s) * 1/(1 – 1/5^s) * …

So when example s = 2, then

1/1² + 2/2² + 1/3² + 2/4² + 1/5² + … = 1/{1 – 2/(2² + 1)} *  1/(1 – 1/3²) * 1/(1 – 1/5²) * …

= 5/3 * 9/8 * 25/24 * …   = 5π²/24.

And again in general terms where s = 2, 4, 6, … and where only a finite number of k₁, k₂, k₃, … terms are given a rational value (with all others = 0) then both the sum over the integers and product over the primes expressions have a value of the form t/π^s (where t is a rational number).  

A very interesting case arises when k₁ = k₂ = k₃ … = 1 for all terms. then

1/1^s + 2/2^s + 2/3^s + 2/4^s + 2/5^s + 4/6^s + 2/7^s + 2/8^s + 2/9^s + 4/10^s + …

 = 1/{1 – 2/(2^s + 1)}* 1/{1 – 2/(3^s + 1)} * 1/{1 – 2/(5^s + 1)} * …

So for example again when s = 2,

1/1² + 2/2² + 2/3² + 2/4² + 2/5² + 4/6² + 2/7² + 2/8² + 2/9² + 4/10² + …

= 1/{1 – 2/(2² + 1)}* 1/{1 – 2/(3² + 1)} * 1/{1 – 2/(5² + 1)} * 1/{1 – 2/(7² + 1)} …

= 5/3 * 10/8 * 26/24 * 50/48 * …

= (5/4 * 4/3) * (10/9 * 9/8) * (26/25 * 25/24) * (50/49 * 49/48) * …

= (4/3 * 9/8 * 25/24 * 49/48 * …) * (5/4 * 10/9 * 26/25 * 50/49 * …)

=  (π²/6) * (15/π²)  = 15/6 = 5/2.

So, interestingly in this particular case where k₁ = k₂ = k₃ … = 1, the value of the sum over the integers and corresponding product over primes expressions is a rational number.

And again this will always be the case where s is an even integer = 2, 4, 6, ...

For example, again where k₁ = k₂ = k₃ … = 1 and s = 4,

1/1⁴ + 2/2⁴ + 2/3⁴ + 2/4⁴ + 2/5⁴ + 4/6⁴ + 2/7⁴ + 2/8⁴ + 2/9⁴ + 4/10⁴ + …

= 1/{1 – 2/(2⁴ + 1)} * 1/{1 – 2/(3⁴ + 1)} * 1/{1 – 2/(5⁴ + 1)} * 1/{1 – 2/(7⁴ + 1)} …

= 17/15 * 82/80 * 626/624 * 2402/2400 * …

= (17/16 * 16/15) * (82/81 * 81/80) * (626/625 * 625/624) * (2402/2401 * 2401/2400) * …

= (16/15 * 81/80 * 625/624 * 2401/2400 * …) * (17/16 * 17/16 * 626/625 * 2402/2401 * …)

= (π⁴/90) * (105/π⁴) = 105/90 = 7/6.

Generalisation of Result

In general terms with reference to the prototype Riemann zeta function, if we choose the primes (in the product over primes expression) using any infinite ordered sequence of integers, then a corresponding sum over the natural numbers expression can thereby be provided.

For example the (infinite) series of triangular numbers i.e. 1, 3, 6, 10, 15, …, provides one such ordered sequence (where we are always enabled to provide the next term in the sequence i.e. n(n + 1)/2, with n = 1, 2, 3, …

So if we now apply this ordering to the primes by choosing the 1^st, 3^rd, 6^th, 10^th 15^th … numbers, then we have 2, 5, 13, 29, 47, …

So the corresponding sum over the integers expression is thereby now based (including 1) on using these primes as sole factors.

Thus

1/1^s + 1/2^s +1/4^s + 1/5^s + 1/8^s + 1/10 ^s …  = 1/(1 – 1/2^s) * 1/(1 – 1/5^s) * 1/(1 – 1/13^s) * …

In fact the prototype Riemann zeta function itself corresponds on an ordering that is equivalent to the denominators of the same zeta function (for s = 1).

So if we now take an ordering based on the denominators of the function for s = 2, it will correspond to the sum of squares 1, 4, 9, 16, 25,…

Thus choosing our primes based on these ordinal rankings we have 2, 7, 23, 53, 97,…

Then the sum over integers expression matching the product over primes expression (using this selection) is based solely (including 1) on numbers based on the factors of such primes.

Thus,

1/1^s + 1/2^s +1/4^s + 1/7^s + 1/8^s + 1/14 ^s …  = 1/(1 – 1/2^s) * 1/(1 – 1/7^s) * 1/(1 – 1/23^s) * …

Of course as always we can then choose to omit any specific prime (or combination of primes) from the RHS expression with corresponding adjustments (based on omission of natural numbers where these are factors) on the LHS.

Therefore in choosing to omit 2 from the RHS (product over primes), we thereby choose to omit all numbers with 2 as a common factor in the LHS (sum over the integers) expression.   

So,

1/1^s + 1/7^s + 1/23^s + 1/49^s + 1/53^s + …  = 1/(1 – 1/7^s) * 1/(1 – 1/23^s) * 1/(1 – 1/53^s) * …

When in the above s = 4 the sum of listed terms for LHS expression = 1.000420367, while the corresponding sum of terms for RHS expression = 1.000420368 (which already shows an extremely close matching).     

Differing Orders of Primes and Natural Numbers (2)

In yesterday's blog entry I dealt with the notion of - what I refer to as - the different orders of primes and natural numbers respectively.

So potentially an unlimited number of orders with respect to both the primes and corresponding natural numbers exist, defined as Order k primes and Order k natural numbers respectively.

And in each case the Order k natural numbers can be represented as a sum over the integers, which can then be matched in corresponding fashion by a corresponding product over the Order k primes.

So again the Order 1 natural numbers (representing all the natural numbers) can be matched with the Order 1 primes (representing all the primes). Thus again,

1/1^s + 1/2^s + 1/3^s + 1/4^s + …    = 1/(1 – 1/2^s) * 1/(1 – 1/3^s) * 1/(1 – 1/5^s) * …

Then the Order 2 natural numbers (representing the products of prime factors which have themselves an ordinal prime ranking) can be matched with the Order 2 primes (i.e. as the primes with an ordinal prime ranking).

So,

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

And then by listing the Order 2 primes in natural number ordinal fashion, we can again pick out those numbers with an ordinal prime ranking that then become the Order 3 primes.

And these in turn are associated with Order 3 natural numbers (as the numbers based on the use of these primes as factors) and so on.

However we can easily derive in each case an alternative set of primes and natural numbers.

So if for example we define the Order 2 primes as those with a prime number ranking, then the alternative set Order 2(a) can be defined as those with non prime rankings.

So the Order 2 (a) primes are thereby 2, 7, 13, 19, 23, …

And the corresponding Order 2(a) natural numbers (based on numbers with these primes as factors, including as always 1) are 1, 2, 4, 7, 8, 13, 14, …

So we can know match the sum of the Order 2(a) natural numbers with the corresponding product of the Order 2(a) primes as follows

1/1^s + 1/2^s + 1/4^s + 1/7^s + … = 1/(1 – 1/2^s) * 1/(1 – 1/7^s) * 1/(1 – 1/13^s) * …

Then by giving the Order 2(a) primes a natural number ranking and then choosing those with a non-prime ordinal ranking we get a new set of Order 3(a) primes, which are associated with a corresponding set of Order 3(a) natural numbers (based on the use of these primes as constituent factors).

So the Order 3(a) primes are 2, 19, 29, 43, …with corresponding Order 3(a) natural numbers 1, 2, 4, 8, 16, 19, …

Thus 1/1^s + 1/2^s + 1/4^s + 1/8^s + …  = 1/(1 – 1/2^s) * 1/(1 – 1/19^s) * 1/(1 – 1/29^s) * …

So in general, just as Order k natural numbers (as sum over the integers) can be associated with corresponding Order k primes (as products over these primes) expressions, likewise Order k(a) natural numbers (as sum over the integers) can be associated with corresponding Order k(a) primes (as products of these primes) expressions.     

And likewise unlimited mixing as between these two approaches is also possible.

So for example as we have seen 3, 5, 11, 17, 31, …,  are the Order 2 primes.

However we could, when listing these primes in a natural number ordinal fashion, choose those with a non-prime ranking. So this new “hybrid” set of primes would thereby be

3, 17, 41, 67, 83, …

And then associated with these primes would be a new set of “hybrid” natural numbers (based on these primes as factors), i.e. 1, 3, 9, 17, 27, 41, 51, …    

We can then match corresponding sum over the integers and product over the primes expressions i.e.

1/1^s + 1/3^s + 1/9^s + 1/17^s + … = 1/(1 – 1/3^s) * 1/(1 – 1/17^s) * 1/(1 – 1/41^s) * …

Differing Orders of Primes and Natural Numbers (1)

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

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, with each successive Order, we eliminate 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 kπ^s (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 + 1/3^s  + 1/5^s + 1/9^s  + 1/11^s + …        = 1/(1 – 1/3^s) * 1/(1 – 1/5^s) * 1/(1 – 1/11^s) * …

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

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/n^s   = ∏1/(1 – 1/p^s)
^{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 2^nd 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. 

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 kπ^s (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. kπ^s 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 1^st and 2^nd 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) = kπ^s , 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) = kπ^s  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.

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 1^st and 2^nd 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.   

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/n^s   = ∏1/(1 – 1/p^s)

^{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 – 2^–s) 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 – 2^–s) ζ(s).

And when again s = 2,

γ₃(2) = (1 – 3^–2) 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 kπ^s (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,

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

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

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 – 2^–s)(1 – 3^–s)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.