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.