Remarkable Relationships (3)

In an earlier posting "Interesting Prime Result", I showed how the (finite) Zeta 2 equation could be used to detect whether a number is prime.

So once the Zeta 2 Equation is defined as:

s^1 + s^2 + s^3 + s^4 +.......s^n = 0

Then dividing by the trivial solution i.e. s = 0 we obtain

1 + s^1 + s^2 + ....... s^(n - 1) = 0.


Nest by letting s = 1 in the expression


y = 1 + s^1 + s^2 + ....... s^(n - 1), by a process of continued differentiation (with respect to s) we showed how to determine whether a number is prime.


However we could also seek to proceed in a complementary direction through obtaining the integral of the same simple expression i.e.

y = 1 + s^1 + s^2 + ....... s^(n - 1).


So ʃy ds = s + (s^2)/2 + (s^3)/3 + (s^4)/4 + .... + (s^n)/n.


Then setting s = 1, we obtain the first n terms in the harmonic series i.e.


1 + 1/2 + 1/3 + 1/4 +..... + 1/n.

Now as n becomes very large the sum of this series approximates very close to log n (which measures the average spread as between prime numbers in the region of n).

For example when n = 1,000,000 the sum of the first n terms = 14.384.

So this provides an approximate measurement of the average spread (or gap) as between prime numbers (in the region of 1,000,000).
Once again this approximation will steadily improve (in relative terms) as n increases.

Thus it is interesting how a simple process of differentiation on this simple (Zeta 2) expression can determine on the one hand whether a number is prime, while the corresponding process of integration can establish the nature of the general distribution among the primes!


If we just concentrate on the first terms of the simple expression we get 1.

Then if we successively integrate with respect to s we obtain s/1!, s^2/2!, s^3/3!,
s^4/4! and so on.

Therefore by adding all these terms we obtain a simple formula for e^s (containing the first n terms of the corresponding infinite expression for e^s).

Of course where s = 1, we approximate the value of e, which becomes ever more accurate as n increases.



Returning to the roots of 1 we showed again how the average value of these roots where n is prime approximates 4/π (especially with respect to the arithmetic mean).

And this approximation steadily improves as the value of n increases.

Once again it is important to bear in mind that we use a reduced linear (1-dimensional) quantitative approach in calculating such values. This means that negative values are treated as positive and imaginary values are treated as real!

However it would also be possible to calculate values using a reduced 2-dimensional quantitative approach.

This entails in effect that negative (as well as positive) values are now considered with however imaginary once again converted to real format.


Now if we attempt to obtain the sum of roots using this approach negative will exactly cancel positive values with result = 0.

However if we obtain the product of such roots a non-trivial result will emerge which in all cases (where n is odd) can be expressed as 1/{(2^(n - 1)/2} in absolute terms.


For example where n = 5, the five roots of 1 (expressed in this 2-dimensional manner) are (to 9 decimal places),


.309016994 + .951056516 = 1.260073510
- .809016994 + .587785252 = - .221231742
.309016994 - .951056516 = - .642039522
- .809016994 - .587785252 = - 1.396802246
1 + 0 = 1


So when we multiply these 5 roots we obtain - .25.
And as in this case (n - 1)/2 = (5 - 1)/2 = 2, the absolute value of .25 = 1/(2^2) conforms to the general formula.

Though the answer in this case is negative, it can also be positive (as for example where n = 9)

Remarkable Relationships (2)

It struck me after I completed the last entry that it was attempting to equate two different types of mean average i.e. the arithmetic and geometric respectively.

Now it is well known that in the non-trivial case where absolute values of numbers differ that the arithmetic mean will always be greater in magnitude that the geometric.

However where the values are all fairly close to 1, the difference in magnitude will be quite small.

For example if we take the five numbers 1, 1.1, 1.2, 1.3 and 1.4 the arithmetic mean = 1.2 and the geometric mean = 1.191596...

Now when we look at the absolute value of the n roots of 1, the numerical value necessarily lies as between 1 and 1.4142... (i.e. the square root of 2).


So we would expect in this case the geometric mean for the n roots to be near to the arithmetic mean (though necessarily smaller).

However what is interesting in the case of the absolute value of these these n roots is that both the arithmetic and geometric means continue to converge more closely on the limiting value 4/π (with however the arithmetic converging more rapidly to this value).


For example when n = 17, the geometric mean = 1.2657221 (with 4/π = 1.27323954...). However when n = 31 the geometric mean = 1.266131209 (which is closer to the limiting value). As we would expect the arithmetic mean for n = 31 is much closer to 4/π (i.e. 1.2729671...).


This convergence towards the limiting value of 4/π, also appears to apply when we confine ourselves merely to the prime numbered roots.


So for example once again when n = 31, there are 11 prime numbered roots i.e. the 2nd, 3rd, 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 29th and 31st respectively.

Now the arithmetic means of these 11 roots = 1.2710048... (which again is very close to 4/π = 1.27323954...). The geometric mean of these same 11 roots = 1.26216 which again reasonably approximates the limiting value).


In fact this highlights a key feature of the linear/circular nature of prime numbers . In other words we can make the important observation that it is in the very nature of prime number behaviour that the average absolute value (both arithmetic and geometric) of all n roots - when n is very large - approximates ever more closely the corresponding arithmetic and geometric averages (entailing the prime numbered roots only).


Thus though though we are concentrating here on the reduced absolute interpretation of root values, they are actually expressive of the linear/ circular behaviour of the prime numbers (with respect to the natural numbers) which is inherent to their very nature.


One remarkable final illustration of this can be given. We have seen repeatedly the importance of 2/π (and counterpart 4/π) in explaining the (reduced) quantitative behaviour of prime numbers (where negative values are treated as positive and imaginary values as real).


In quantitative terms 2/π = i/log i.

Now if we again attempt the same reduced (imaginary to real) quantitative transformation on this relationship we obtain 1/log 1 i.e. 1/0.

Now from a quantitative perspective, this result is somewhat meaningless. However if we now switch back to qualitative interpretation 1/0 implies the relationship as between linear and circular meaning (i.e. quantitative and qualitative). And this precisely defines the relationship as between the prime and natural numbers (in qualitative terms)!