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.

No comments:

Post a Comment