Stunning Accuracy!

I have mentioned several times before the formula for calculating the frequency of the non-trivial zeros up to any given height on the imaginary number line.

This can be expressed for convenience as:

t/2π{(log t/2π) – 1} where t represents the required height on the imaginary scale.

This in fact represents the “circular” version of the corresponding simple "linear" formula for calculating the average number of primes up to t on the real number scale,

i.e. t/log t.

However the latter formula – though progressively improving as a predictor of the number of primes in relative percentage terms as t increases – is hopelessly inaccurate as a predictor in absolute terms.

For example the actual number of primes up to t = 1,000,000 = 78,498.

However our simple formula predicts 72,382 (which undershoots the correct result by 6,116).

Though the formula will steadily improve as a predictor in percentage terms as t increases, the absolute deviation from the correct number of primes likewise also increases (at least for values of t that can be feasibly calculated)!

However by contrast the corresponding formula for predicting the number of non-trivial zeros is stunningly accurate, not only in relative percentage, but also in absolute terms.

Due to Andrew Odlyzko extensive tables of these non-trivial zeros have now been made available.

The accuracy  of earlier calculations with the formula impressed me greatly. However I did not realise until discovering these tables, how stunningly accurate is the formula apparently as a means of predicting the absolute number of these zeros to any required height.

The formula in fact seems to under-predict results on average by 1.

So the slightly modified version that I suggest is

t/2π{(log t/2π) – 1} + 1.

For example the number of zeros up to 100 is 29.

When we use this formula we obtain 29.1273… (which rounded to the nearest integer gives the exact answer).

Then the number of zeros up to 1000 is 649.

Again using the formula we obtain 648.7412 (which rounded gives the exact answer).

Using Odlyzko’s tables the 100,000^th zeros occurs at t = 74920.827… with 99,998 zeros up to 74920.

Using the formula with t = 74920, the number of zeros =  99998.2939… (which again rounded is the exact answer).

There are 10^12 + 1 zeros up to t = 267,653,395,649.

Using the formula for this value of t, the number of zeros = 1,000,000,000,000.616 (which once again rounded to the nearest integer gives the exact answer).

There are 10^21 + 5 zeros up to t = 144,176,897,509,546,973,539.

Using the formula the number of zeros = 1,000,000,000,000,000,000,004.02568… (which, when rounded is just 1 less than the correct answer). The slight underestimate here is due to the higher than average degree of clustering of zeros as between 8 and 9 with respect to the last digit of t!

Finally to illustrate there are 10^22 + 3 zeros up to t = 1,370,919,909,931,995,308,227.

Using the formula for this value of t, the calculated number of zeros = 10,000,000,000,000,000,000,002.3376 (which when rounded is again just 1 less than the correct answer. Once again the explanation is the higher than average degree of clustering of zeros in the region of the last digit of t.

The accuracy of these predictions of the frequency of the imaginary part of non-trivial zeros is simply stunning! Indeed we could conclude that on average i.e. where typical levels of clustering are in evidence in the region of the number t in question that – when rounded – the exact answer will emerge.

This contrast therefore sharply with the corresponding prediction of primes, with the absolute deviation (from the actual number of primes) tending to increase (as t increases) to very large numbers.

Indeed this throws fascinating light on the distinction as between the primes and corresponding non-trivial zeros.

The primes are characterised at an individual level by their uniquely independent features while equally being characterised at a general level by their universal common behaviour (with respect to the natural numbers).

However this implies an inevitable clash as between the accuracy of their prediction in relative and absolute terms.

The common universal pattern tends to dominate for large values of n leading to ever more accurate predictions of their frequency (in relative percentage terms).

However the individual independent features of the primes likewise tend to accumulate - at a slower rate - with larger n leading to growing deviations from their actual frequency (in absolute number terms).

I have frequently stated that the non-trivial zeros represent the shadow system to the primes.

So the very nature of these numbers is that they represent the reconciliation of both the analytic (independent) and holistic (interdependent) aspects of the primes with respect to the natural numbers.  Expressed another way, they represent the continual smoothing out of the deviations arising from the distinctive behaviour of the individual and universal features respectively of the primes.

Now this process - as it attempts to bridge both the discrete and continuous aspect of the number system - can never be perfect. Therefore we would always expect some small deviations to remain in locally confined areas of the number system.

So we cannot therefore guarantee absolutely precise predictions from this simple formula of the frequency of non-trivial zeros. However because their frequency increases as t increases, deviations (of the individual from the general behaviour of the primes) become confined to ever smaller local areas of the number system.

We therefore would not expect the absolute deviation (where it arises) from actual results to grow in any systematic manner, as by definition these zeros continually strive to reconcile the individual (analytic) with the universal (holistic) features of the primes.  

Put yet another way - as I have repeatedly highlighted in these blog entries - both the cardinal and ordinal features of the primes are uniquely determined according to two distinct aspects of the number system (Type 1 and Type 2 respectively).

The non-trivial zeros thereby represent the reconciliation of both  Type 1 and Type 2, which occurs in the combined dynamic complementary interaction of both aspects i.e. Type 3.

This highlights yet again the futility of the conventional mathematical approach in attempting to understand prime number behaviour in mere Type 1 terms!