Complementary L1 and L2 Functions

In the last blog entry, I have been concentrating attention on the complementary dynamic relative nature of both the infinite sum over the natural numbers and corresponding infinite sum over the primes expressions (which characterise all L-functions).

Therefore what is interpreted in an analytic (quantitative) manner with respect to the right-hand expression (i.e. product over primes) should be interpreted in a corresponding holistic (qualitative) manner with respect to the left-hand expression (i.e. sum over natural numbers).

Equally in reverse, what is interpreted in a holistic (qualitative) manner with respect to the right-hand expression should be interpreted in a corresponding analytic (quantitative) manner with respect to the left-hand expression.

Thus all individual terms (and collection of terms) with respect to both expressions can thereby be given twin analytic (quantitative) and holistic (qualitative) interpretations that interact with each other in a dynamic complementary manner.

This dynamic complementarity can also be illustrated in another closely related manner.

When we look at the Riemann zeta function which expresses a collective infinite sum over the natural numbers i.e.

1^{– s} + 2^{– s} + 3^{– s} + 4^{– s}  + …,

the base aspect of number varies over the positive integers, whereas the corresponding dimensional aspect i.e.  – s remains fixed.

When we now look at each individual term of the corresponding product over primes expression, it is given as 1/(1 – 1/p^s).

However this can be written as an infinite sum,

= 1 + (1/p^s)¹  + (1/p^s)²  + (1/p^s)³  + (1/p^s)⁴  + …

So now here in an inverse complementary manner the dimensional aspect of number varies over the positive integers, whereas the base aspect remains fixed (as 1/p^s).

Thus again when s = 2,

the collective sum over the natural numbers

= 1/1² + 1/2² + 1/3² + 1/4² + …  = 1 + 1/4 + 1/9 + 1/16 + …   = π²/6.

Then for the first individual terms of the product over primes expression (with p = 2),

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

1 + 1/4 + 1/16 + 1/64 + …   = 4/3.

In an earlier blog entry, Dynamic Perspective on Addition and Multiplication, I explained how the dynamic complementary interaction as between addition and multiplication is necessary to explain how the meaning of number itself switches as between base and dimensional aspects (representing in turn the interaction of both quantitative and qualitative aspects).

And now here we see this beautifully exemplified with respect to both the collective (infinite) sum over natural numbers expression and each individual term of the (infinite) product over the primes.

It is because of this dynamic complementarity that I refer to the Riemann zeta function (collective sum over the natural numbers) as the Zeta 1 function and the alternative infinite function, for each individual term in the product over primes expression, as the Zeta 2 function respectively.

Thus the Zeta 1 and the Zeta 2 functions act in a dynamic complementary manner and this in turn is true for all L functions.

So if we refer to the (collective) infinite sum over natural numbers expression as the L1 function, then each individual term in the corresponding infinite product over primes expression operates in a complementary fashion (with respect to base and dimensional aspects) and can be referred to as the L2 function.