Sunday, November 14, 2010

Thr Pi Connection

Euler also demonstrated another remarkable connection as between his Zeta function and the value of pi.

So whenever (representing the dimension) in the function is a positive even integer, then the resulting value can be expressed as pi (to the power of s) * by a rational no.

So in the simplest - and best known - case when s = 2,

∑[1/n^2] = ∏{p^2/(p^2 – 1)} = (pi^2)/6

One of the interesting implications of this result is that it provides another means of proving the infinitude of primes (i.e. in the accepted reduced nature of the infinite).

For if the the no. of primes was finite then the product formula (involving the primes) would ensure a rational value.

However because the actual answer is irrational (involving pi), then this implies that the no. of primes must be infinite.

However the deeper qualitative implications of this result are not properly appreciated.

Again no matter how many finite terms are included in the product formula, a rational result will ensue. Therefore the fact that the result is irrational, implies that an additional qualitative aspect of understanding is required.

Now some infinite series in the limit of an infinite process - again accepting the reduced conventional appreciation of the infinite - do result in a rational finite answer.

For example if we sum the series 1 + 1/2 + 1/4 + 1/8 +......,

the actual sum for any finite no. of terms will be rational. However the limiting value for an infinite no. of terms will also be rational (i.e. 2).

However one reason why this infinite series does not lead to an irrational value is the fact that each successive term can itself be expressed as a rational fraction of the previous term. So just as rational number can be expressed in decimal form with a consistent repeating sequence of digits, likewise if successive terms in a series are related in a similar manner (through the application of a consistent rational operation) then a rational limit will result.

However clearly in the case of series entailing the prime numbers, this is not in fact the case. Rather, as I have stated before the actual location of each prime number intimately depends on the overall holistic relationship of the primes to the natural numbers.

And, again the key point is that this holistic relationship is qualitatively of a different nature as it relates to the potential infinite nature of these numbers whereas rational interpretation relates properly to actual finite type considerations.

So once again the key limitation of Conventional Mathematics is that it can only attempt to deal with potential infinite notions - properly relating to circular paradoxical type appreciation - in a grossly reduced manner. Here they are treated as an extension of the finite (so as to become amenable to an unambiguous linear type logic).

As we know the constant pi (which is irrational and transcendent in nature) pertains quantitatively to the relationship of the (circular) circumference to its (linear) diameter.

Likewise in holistic mathematical terms, qualitative understanding (that is irrational and transcendental) pertains to the corresponding relationship as between (pure) circular and linear type interpretation.

This thereby implies that - correctly understood in an appropriate qualitative manner - interpretation of the Zeta function (for even values of s) implies the pure relationship as between linear and qualitative type notions that is expressed in an indirect rational manner.

In other words - when appropriately understood - we then realise that the very nature of prime numbers entails a pure relationship as between actual finite notions (in the precise identity of specific prime numbers)and potential infinite notions (in the general distribution of the primes among the natural number system).

However there is another fascinating connection in these pi expressions and the nature of the primes.

If we confine ourselves to the rational nos. by which the pi expressions must be multiplied, then the denominators of such expressions bear a very important relationship to the primes.

So, when s = 2n (where n = 1, 2, 3, 4,...)

then, where 2n represents an integral power of 2, the denominator of the rational number part will represent a product of all prime numbers (in various combinations) up to 2n + 1 and only these primes.

In all other cases, the denominator of the rational part will represent the product of all prime numbers - in varied combinations - from 3 to 2n + 1(and only these primes).

For example when s = 2n = 4, this represents an integral power of 2. The denominator of the rational part = 90 = 2 * 3^2 * 5 which represents all primes from 2 to 2n + 1(i.e. 5).

Then when s = 2n = 6, this does not represent an integral power of 2.
The denominator of the rational part = 945 = 3^3 * 5 * 7 which represents all primes from 3 to 2n + 1 (i.e. 7).

I will just give one more example to illustrate

Wne s = 2n (i.e. n = 8) = 16, s can again be expressed as 2 (raised to a positive integral power) i.e. 2^4.

The denominator here of of pi^16 = 325641566250.

And 325641566250 = 2 * 3^7 * 5^4 * 7^2 * 11 * 13 * 17. So we can see here how all the prime numbers from 2 to 2n + 1 i.e. 17, are included as factors in the denominator (and only these primes).

No comments:

Post a Comment