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,

1/1

^{2}+ 1/2^{2 }+ 1/3^{2 }+ 1/4^{2 }+ … = 1/(1 – 1/2^{2}) * 1/(1 – 1/3^{2}) * 1/(1 – 1/5^{2}) * …
i.e. 1 +
1/4 + 1/9 + 1/16 + … = 4/3 * 9/8 * 25/24 * … = π

^{2}/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 1/1

^{2}+ 1/3^{2 }+ 1/5^{2 }+ 1/7^{2 }+ … = 1/(1 – 1/3^{2}) * 1/(1 – 1/5^{2}) * 1/(1 – 1/7^{2}) * …
And since
we have removed 4/3 from the previous RHS expression, the value of this new
expression = (π

^{2}/6)/(4/3) = (π^{2 }* 3)/24 = π^{2}/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) * (π

^{2}/6) = π^{2}/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 1/1

^{2}+ 1/2^{2 }+ 1/4^{2 }+ 1/5^{2 }+ … = 1/(1 – 1/2^{2}) * 1/(1 – 1/5^{2}) * 1/(1 – 1/7^{2}) * …
And as the
eliminated term in the product over primes expression is 9/8, then the
corresponding value of this new L-function = (π

^{2}/6)/(9/8) = (π^{2 }* 8)/54 = 4π^{2}/27.
So this now
leads to a new Lambda Function” which I will denote as γ

_{3}(s).
Thus γ

_{3}(s) = (1 – 2^{–}^{s}) ζ(s).
And when again
s = 2,

γ

_{3}(2) = (1 – 3^{–}^{2}) ζ(2) = 8/9 * π^{2}/6 = 4π^{2}/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,

1/1

^{2}+ 1/5^{2 }+ 1/7^{2 }+ 1/11^{2 }+ … = 1/(1 – 1/5^{2}) * 1/(1 – 1/7^{2}) * 1/(1 – 1/11^{2}) * …
And the
value of this new L-function = π

^{2}/6 * 3/4 * 8/9 = π^{2}/9.
And if we
refer to this new Lambda Function” as γ

_{23}(s), then
γ

_{23}(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.

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