Monday, January 30, 2012

Further Thoughts on Non-Trivial Zeros

It struck me forcibly over the weekend that the relationship between the distribution of prime numbers and the corresponding distribution of non-trivial zeros is even more intimate than I had realised.

Therefore it is very easy to move from a knowledge of the distribution with respect to one distribution to corresponding knowledge with respect to the other.

For example the average spread of prime number in the region of n is given simply as log n.

Therefore for example in the region of hundred the average spread (or gap) as between prime numbers is 4.605 (approx).

Now if we let n = t/2pi, we can correspondingly obtain the average gap as between the trivial non-zeros in the region of t = n*2pi = 628.3(approx).

This gap is given by the formula 2pi/{log (t/2pi)} = 6.283/4.605 = 1.3644 (approx).

So we can say therefore that the average spread as between non-trivial zeros in the region of 628 = 1.364 (approx).

So to obtain this latter answer (in the region of t) we simply divided 2pi by the average spread as between primes (in the region of n)!

We could of course equally calculate the average spread as between primes (for any value of n) from the spread as between non-trivial zeros (for the corresponding value of t).

As we have seen we can calculate the change in the average spread as between primes as n increases to n + 1 by the simple expression 1/n.

Therefore by we would expect the average spread as between primes to increase by 1/100 as we move from 100 to 101 (more correctly from 99.5 to 100.5).

A corresponding expression can be obtained through differentiation with respect to t in the original expression for calculating the gap as between non-trivial zeros,

= -(4*pi^2)/{log [(t/2pi)^2]*t}. This is negative as the average gap between non-trivial zeros decreases as the value of t increases!

So once again where n = 100 = t/2pi, it works out as -.00296 (approx)

Therefore in moving from t to t + 1, the average gap as between non-trivial zeros declines by .003 (approx).

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