Another Interesting Generalisation (1)

It just struck recently that the sum over the integers and product over the primes expressions, with respect to the Riemann zeta function, represent a special case of a more comprehensive relationship, which can be expressed in general terms as follows:

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, … are rational numbers which can be either positive or negative.


So the standard equation represents the situation where the values for  k₁, k₂, k₃, … = 0_. 

It can be seen here it is the distinct prime factors of a number that determines the nature of the sum over the integers expression.

So for example, 6 is the 1st composite natural number with two distinct prime factors. Therefore as we can see from the new formula we must here thereby multiply 1 + k₁ (associated with 2) by 1 + k₂ (associated with 3) respectively.

One very simple case with respect to this general formula is for k₁ = 1 and k₂, k₃, k₄, … = 0

Then,

1/1^s + 2/2^s + 1/3^s + 2/4^s + 1/5^s + …  =  1/{1 – 2/(2^s + 1)} *  1/(1 – 1/3^s) * 1/(1 – 1/5^s) * …

So when example s = 2, then

1/1² + 2/2² + 1/3² + 2/4² + 1/5² + … = 1/{1 – 2/(2² + 1)} *  1/(1 – 1/3²) * 1/(1 – 1/5²) * …

= 5/3 * 9/8 * 25/24 * …   = 5π²/24.

And again in general terms where s = 2, 4, 6, … and where only a finite number of k₁, k₂, k₃, … terms are given a rational value (with all others = 0) then both the sum over the integers and product over the primes expressions have a value of the form t/π^s (where t is a rational number).  

A very interesting case arises when k₁ = k₂ = k₃ … = 1 for all terms. then

1/1^s + 2/2^s + 2/3^s + 2/4^s + 2/5^s + 4/6^s + 2/7^s + 2/8^s + 2/9^s + 4/10^s + …

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

So for example again when s = 2,

1/1² + 2/2² + 2/3² + 2/4² + 2/5² + 4/6² + 2/7² + 2/8² + 2/9² + 4/10² + …

= 1/{1 – 2/(2² + 1)}* 1/{1 – 2/(3² + 1)} * 1/{1 – 2/(5² + 1)} * 1/{1 – 2/(7² + 1)} …

= 5/3 * 10/8 * 26/24 * 50/48 * …

= (5/4 * 4/3) * (10/9 * 9/8) * (26/25 * 25/24) * (50/49 * 49/48) * …

= (4/3 * 9/8 * 25/24 * 49/48 * …) * (5/4 * 10/9 * 26/25 * 50/49 * …)

=  (π²/6) * (15/π²)  = 15/6 = 5/2.

So, interestingly in this particular case where k₁ = k₂ = k₃ … = 1, the value of the sum over the integers and corresponding product over primes expressions is a rational number.

And again this will always be the case where s is an even integer = 2, 4, 6, ...

For example, again where k₁ = k₂ = k₃ … = 1 and s = 4,

1/1⁴ + 2/2⁴ + 2/3⁴ + 2/4⁴ + 2/5⁴ + 4/6⁴ + 2/7⁴ + 2/8⁴ + 2/9⁴ + 4/10⁴ + …

= 1/{1 – 2/(2⁴ + 1)} * 1/{1 – 2/(3⁴ + 1)} * 1/{1 – 2/(5⁴ + 1)} * 1/{1 – 2/(7⁴ + 1)} …

= 17/15 * 82/80 * 626/624 * 2402/2400 * …

= (17/16 * 16/15) * (82/81 * 81/80) * (626/625 * 625/624) * (2402/2401 * 2401/2400) * …

= (16/15 * 81/80 * 625/624 * 2401/2400 * …) * (17/16 * 17/16 * 626/625 * 2402/2401 * …)

= (π⁴/90) * (105/π⁴) = 105/90 = 7/6.