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 – 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 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 – 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.