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.