## Friday, July 20, 2018

### 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:

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

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

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

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/πs (where t is a rational number).

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

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

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

=  2/6) * (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.