Monday, January 9, 2012

Frequency of Non-Trivial Zeros

As is well known the non-trivial zeros of the Riemann Zeta Function for dimensional value s are given by s = .5 + it (and its conjugate expression s = .5 - it).

Now the first of these zeros of s occurs where the value of t = 14.134725 and the second where t = 21.022040!

The spread (or gap) as between the imaginary part of these first two zeros is approximately 7! However this gap steadily falls as the value of t increases so that when t for example = 100 the average gap is just a little over 2!

In fact a very interesting formula exists to calculate this average spread between zeros (for different values of t) i.e. 2π/log(t/2π) using the natural log based on e.

Thus when t = 100, the average spread = 6.281385/log 15.915494 = 2.27 (approx).

2π represents the circumference of the circle of unit radius.

If we replace in the formula this circular value for the circumference by the corresponding linear value of its radius we obtain 1/log t.

1/log t represents the probability that the number t is prime!

If we obtain the inverse of this expression then log t represents the average gap (or spread) as between primes in the region of t!

So log 100 (= 4.6 approx) for example represents the average spread as between primes in the region of 100.

There appears to be a very close relationship as between both sets of results.

In the first instance the relationship as between real and imaginary values is as between linear and circular. Thus as the spread between non-trivial zeros relates to the imaginary part (of a dimensional value) whereas the corresponding spread as between prime numbers represents a real dimensional value (i.e. the value that e must be raised by to obtain the number t is question) we would expect the relationship as between the two types of spreads in turn to be circular and linear with respect to each other.

Also the relationship as between linear and circular is of a reciprocal nature. For example the prime number 7 is a good example of a linear number (with no sub factors). However its reciprocal 1/7 = .142857 (recurring) is a prime example of a number with a circular structure!

One would expect the spread as between primes and that as between the non-trivial zeros to be the inverse of each other. Thus though the primes thin out in frequency as we ascend the real number scale, the frequency of non-trivial zeros increases as we ascend the imaginary number scale.

What this in fact entails is that as the number of primes thins out, the corresponding number of composite natural numbers proportionately increases (with a greater average number of prime factors).

The connections between both estimates (i.e. for average spread as between non-trivial zeros and the average spread as between primes) is so close that one would seem to imply the other. They are like two sides of the same coin (as it were).

A fascinating corollary of this would imply that just as it is possible to correct the general estimate of prime number distribution with a set of spiral deviations (based on the non-trivial zeros) to get a precise estimate, likewise in principle it should be possible to correct the general estimate of frequency of occurrence of non-trivial zeros (up to a certain number t) with a corresponding set of deviations based on the primes to get the precise number of zeros.

In fact the general estimate of occurence of zeros up to t is given by

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

Ignoring the final error term this would predict that the number of zeros up to t = 100 = 15.9155*2.767293 - 15.9155 = 15.9155*1.767293 = 28.127.

This equates very well for the actual number of zeros = 29.

Recently I have realised that the final O log term which would be used to correct the deviation between predicted and actual zeros could involve a corollary inverse statement to the Riemann Hypothesis in the recognition that all prime numbers are ultimately based on 2!

Incidentally it is interesting when we substitute 2iπ (rather than 2π) in the original formula for calculation of spread between the non trivial zeros.

Then we get 2iπ/log (t/2iπ)

Now 2iπ = log 1.

Therefore we can rewrite formula as log 1/{log(t/log 1)}.

Once again if we replace log 1 with 1, we get 1/log t (i.e. the probability that t is prime).

Also in the latter formula for calculation of frequency of zeros up to the number t (on imaginary scale) if we replace the (circular) circumference of unit circle 2π by its corresponding (linear) radius the RHS of expression

= t(log t) - t = t(log t - 1).

The calculation of frequency of primes up to t is most simply given as t/log t.

However the accuracy of the estimate can be improved for lower t) by substituting log t by log t - 1.

For example if t = 1,000,000 the estimate of the number of primes up to this number
using t/log t = 72,382 (as against true estimate of 78,498)

However if we use t/(log t - 1), we get estimate of 78,030!

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