More on Riemann Generalisation (1)

In an earlier entry “Intertwining L1and L2 Functions” I showed that,

(1/1² + 1/2²  + 1/3² + 1/4² + …)²/(1/1⁴ + 1/2⁴  + 1/3⁴ + 1/4⁴ + …)  

= (π²/6)²/(π⁴/90)  = 90/36 = 5/2.

And  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

This in turn makes use of our generalised Riemann zeta function formula,

1/1^s + (1 + k₁)/2^s + (1 + k₂)/3^s + (1 + k₁)/4^s + (1 + k₃)/5^s + {(1 + k₁)(1 + k₂)}/6^s + …

= 1/{1 – (1 + k₁)/(2^s + k₁)} * 1/{1 – (1 + k₂)/(3^s + k₂)} * 1/{1 – (1 + k₃)/(5^s + k₃)} * …,

 

where k₁, k₂, k₃, …   = 1.

So ζ(2)²/ζ(4) = 1/1² + 2/2² + 2/3² + 2/4² + 2/5² + 4/6² + 2/7² + 2/8² + … = 5/2

And the result can be generalised so that when s is a positive even integer,

(1/1^s + 1/2^s  + 1/3^s + 1/4^s + …)²/(1/1^2s + 1/2^2s  + 1/3^2s + 1/4^2s + …)  = k (where k is a rational number).

And this can be expressed as an L1 series (with exponent s) i.e.

1/1^s + 2/2^s  + 2/3^s + 2/4^s + 2/5^s + 4/6^s + 2/7^s + …

Here the value of numerator is determined by the number of distinct prime factors contained by each corresponding number in the denominator with the numerator = 2^t, where t represents the number of distinct prime factors.

So 2, 3, 4 and 5 respectively each contain one distinct prime factor. So the numerator is 2¹ = 2.

However 6 then contains two distinct prime factors so the numerator in this term = 2² = 4.

And the corresponding general product over primes expression for this new L-series is,

1/{1 – 2/(1 + 2^s)} * 1/{1 – 2/(1 + 3^s)} * 1/{1 – 2/(1 + 5^s)} * …

Thus to illustrate once again when s = 4, we have

(1/1⁴ + 1/2⁴  + 1/3⁴ + 1/4⁴ + …)²/(1/1⁸ + 1/2⁸  + 1/3⁸ + 1/4⁸ + …)  = (π⁴/90)²/(π⁸/9450)

= (π⁸/8100)/(π⁸/9450)  = 9450/8100 = 7/6

And this is represented by the new L-series (with exponent 4) i.e.

1/1⁴ + 2/2⁴  + 2/3⁴ + 2/4⁴ + 2/5⁴ + 4/6⁴ + 2/7⁴ + …,

with a corresponding product over primes expression,

1/{1 – 2/(1 + 2⁴)} * 1/{1 – 2/(1 + 3⁴)} * 1/{1 – 2/(1 + 5⁴)} * …

= 1/(1 – 2/17) * (1 – 2/82) * (1 – 2/626) * …

= 17/15 * 82/80 * 626/624 * …

And again each of the individual terms (for both the sum over the integers and product over the primes expressions) can then be expressed by a corresponding L2 series.

For example the first term of the product over primes expression i.e. 17/15

= 1 + (2/17)¹ + (2/17)²  + (2/17)³  + …  = 17/15

And once more an unlimited number of possible variations can be made on this basic result through removing certain prime numbers (in the product over primes expressions) and then the corresponding natural numbers where these are factors in the sum over the integers expression.

So for example if we remove 2 (in product over primes expression) and then all corresponding even numbers (in the sum over integers expression) we have

 (1/1⁴ + 1/3⁴  + 1/5⁴ + 1/7⁴ + …)²/(1/1⁸ + 1/3⁸  + 1/5⁸ + 1/7⁸ + …)  = (π⁴/96)²/(255π⁸/9450 * 256) 

= 35/34

And this is represented by the new L-series expression,

1/1⁴ + 2/3⁴  + 2/5⁴ + 2/7⁴ + 2/9⁴ + 2/11⁴ + 2/13⁴ + 4/15⁴ + …,

with a corresponding product over primes expression

1/{1 – 2/(1 + 3⁴)} * 1/{1 – 2/(1 + 5⁴)} * 1/{1 – 2/(1 + 7⁴)} * …

= 1/(1 – 2/82) * (1 – 2/626) * (1 – 2/2402) * …

= 82/80 * 626/624 *  2402/2400 * …  = 35/34

And just as this new L1 series represents a quotient of the two original L1 series,

in complementary fashion, each individual terms can be expressed as a product of two original L2 series

So for example 82/80 = 82/81 * 81/80

= [1 + {1/1 + 3⁴}¹ + {1/1 + 3⁴}² + {1/1 + 3⁴}³ + …] * [1 + (1/3⁴}¹ + (1/ 3⁴)² + (1/3⁴)³ + …].