## Monday, July 23, 2018

### Another Interesting Generalisation (2)

In the last entry, I mentioned the new general formula, which can be used to extend relationships as between sum over integers and product over primes expressions with respect to the Riemann zeta function i.e.

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.

In that entry, I demonstrated the - surely - interesting fact that when s is 2, 4, 6, …

and k1 = k2 = k3,   = 1, that the value of the two expressions (sum over the integers and product over the primes respectively) is a rational number.

In this case there is a clear relationship as between the values of the standard zeta function for s and 2s respectively.

So - again when s is a positive even integer - the value of the zeta expression (for both sum over the integers and product over the primes expression) is t1πs (where t1 is a rational number).

Likewise the value of the expression when the dimension is 2s is t2π2s (where t2 is a rational number).

Let the ratio t2/t1  = t3.

Then the value of the extended zeta function where k1 = k2 = k3,   = 1, is t3/t1 i.e. t2/(t1)2.

So when s = 2, ζ(2) = π2/6 and when s = 4, ζ(4) = π4/90.

So t1 = 6 and t2  = 90.

Therefore t2/(t1)2  = 90/36 = 5/2

And this is the value of the extended expression where k1 = k2 = k3,   = 1.

Another interesting - if trivial - case arises when k1 = k2 = k3,   = – 1.

This leads to the elimination on both sides of all terms entailing k so that we are left with the identity 1 = 1.

This is useful to remind us that 1 precedes all subsequent relationships entailing prime factors. So 1 is not obtained from these relationships but rather serves as a necessary precondition for their use.

Thus internally the very notion of a prime entails unit components which then have both a quantitative identity as (independent of each other) and a qualitative identity as (interdependent with other) respectively.

So far in the use of the extended general formula we have shown how to generate expressions where the numerator  > 1.

However by picking the value of k1, k2, k3, …, appropriately we can likewise generate expressions where the denominator - rather than the numerator term -  increases.

For example when k1 = – 1/2, k2 = – 2/3, k3 = – 4/5, …, we have

1/1s + (1 – 1/2)/2s + (1 – 2/3)/3s + (1 – 1/2)/4s + (1 – 4/5)/5s + {(1 – 1/2)(1 – 2/3)}/6s + …

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

i.e.

1/1s + 1/(2.2s) + 1/(3.3s) + 1/(2.4s) + 1/(5.5s) + 1/(6.6s) + …

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

So for example, when s = 2

1/12 + 1/(2.22) + 1/(3.32) + 1/(2.42) + 1/(5.52) + 1/(6.62) + …

= 1/{1 – [(1/2)/(22 – 1/2)]} * 1/{1 – [(1/3)/(32 – 2/3)]} * 1/{1 – [(1/5)/(52 – 4/5]} * …,

= 1/(1 – 1/7) * 1/(1 – 1/25) * 1/(1 – 1/121) * …= 7/6 * 25/24 * 121/120 * …

And likewise k1, k2, k3, …, can be chosen so that both numerator and denominator are multiples of values that occur in the standard case.