Addendum on Frequency of Non-Trivial Zeros

In the last post I mentioned the formula (which is surprisingly accurate) for the calculation of the number of non-trivial zeros up to a certain number.

This formula is

N(t) = t/2π*log(t/2π) - t/2π + O(log t)

Now again ignoring the final error term we can use this to calculate that the number of non-trivial zeros - for example - up to t = 100, is 28.127 (as against the correct prediction of 29).

As the calculation of the change in the spread as between primes in the region n (i.e. as n increases by 1) is given by the remarkably simple estimate of 1/n, it would therefore be very interesting to find a similar type estimate in the change of the spread as between the non-trivial zeros in the region of t.

In other words as t increases by 1 to t + 1, what is the change in the frequency of occurrence of non-trivial zeros?

Using my somewhat rusty application of rules in differentiating the formula at the top with respect to t, I came up with the estimate 1/2π(log t/2π).

In other words as t increases to t + 1, the change in the frequency of occurrence of the non-trivial zeros is given as 1/2π(log t/2π).


And this formula is simply the inverse of that which we have devised for calculation of the average spread in the frequency of non-trivial zeros.

So in fact, as we already calculated if the average gap as between trivial zeros in the region of 100 = 2.27, then the change in the frequency of non-trivial zeros per unit change in t is given as the inverse 1/2.27 = .44 (approx).


In particular this would imply that if we find the value of t for which the average spread as between non-trivial zeros = 1, then likewise the change in frequency of such zeros (in increasing from t to t + 1) will also be equal to 1.

Now earlier in the last post we saw that the formula for calculation of the average gap or spread as between non-trivial zeros is given as 2π/(log t/2π).

So for this value of t, 2π/(log t/2π) = 1
therefore 2π = log (t/2π)

so e^2π = t/2π {or remarkably 1^(-i) = t/2π}


So t works out at 3364.6 (approx) = 2π{1^(-i)} or 2π/(1^i)

So when t = 3365 (rounded to nearest integer) both the average gap in t for each new non-trivial zero and the change in the total frequency of non-trivial zeros (as t increases by 1) is approximately identical.


And finally, one must comment again on the similarity as between formulae with respect to behaviour of the primes on one hand and the behaviour of non-trivial zeros on the other. In each case the resemblance can be clearly shown through a linear to circular (or circular to linear) transformation on the one hand and then a simple inversion.

So the formula we worked out to calculate the change in frequency of the non-trivial zeros (as t increases to t + 1) is given as 1/2π(log t/2π).

Once again when we substitute the circular circumference of the unit circle 2π with its linear radius we obtain log t.

Now log t measures the spread as between primes in the region of t (where t here is measured on the real number scale), whereas its differential 1/t measures the change in this spread (as t increases to t + 1).


And once again I personally find it truly astonishing that the value for t (where the average gap between non-trivial zeros = 1 (which equally is the value where the frequency on such zeros increases by 1 as t goes to t + 1) is given as 2π/(1^i).

This highlights for me the direct relationship here as between linear and circular notions (and quantitative and qualitative) in the understanding of the primes!

However because Conventional Mathematics only deals in the quantitative aspects of linear and circular notions (while qualitatively locked in a merely linear 1-dimensional approach) it fails to recognise the true beauty of the primes which relates to this direct link as between quantitative and qualitative notions.