Monday, April 7, 2014

Simple Estimate of Frequency of Riemann Zeros (1)

I have mentioned many times before how the Riemann (non-trivial) zeros of the zeta (i.e. Zeta 1) Function bear a complementary dynamic relationship with the prime numbers.

Thus from this context, the primes represent the "masculine" extreme as the independent aspect representing the building blocks of the natural number system. This is demonstrated by the fact that primes - by definition - have no factors (other than themselves and 1). The non-trivial zeros by contrast thereby represent the corresponding "feminine" extreme expressing the interdependence of the primes with the number system.

I have suggested before that the interdependent aspect is represented by the remaining composite natural numbers (that reflect a unique combination of prime factors).

Indeed I had earlier attempted to predict the frequency of the non-trivial zeros by a process of summing up the number of prime factors associated with each composite number.

However it became quickly apparent that such an approach would significantly underestimate the actual frequency.

Only last week, I realised that a modified approach based on counting the total number of factors associated with each composite number (rather than just prime factors) would offer greater promise of being on the right track.

And to my pleasant surprise I have found that it does indeed offer a very simple - and surprisingly accurate - way of measuring the frequency of the Riemann zeros.

However there is a slight variation here from what are termed the proper factors of a number.

For example a perfect number such as 6 is perfect as the sum of its proper factors (i.e. 1, 2 and 3) = 6!

Now there is something arbitrary about this definition of factors. Whereas 2 and 3 in this context will be readily accepted as "true" factors, one might indeed query the inclusion of 1. As all numbers by definition are divisible by 1, this could therefore be viewed as a "trivial" case which therefore should be omitted.

Also if 1 is included why not also 6 as clearly 6 is equally divisible by 6 and 1!
However very interesting results follow from defining the proper factors in this way. So the important thing is to be consistent in applying one's initial definition.

In the context of counting factors for the purpose of calculating the frequency of trivial zeros, I use a slightly different approach.

Here, if the number is prime then no factors are considered (which excludes therefore consideration of 1 and the prime number itself).

However where the number is composite, this number is included as a factor plus all other factors (except 1).
So for example once again in the in the case 6 (which is composite), 6 is now included as a factor, whereas in the case of 7 (which is prime) 7 would be excluded. The rationale for this is that 6 itself has sub factors whereas 7 does not!

So in the case of 6 we therefore have 3 factors (i.e. 2, 3 and 6).

To illustrate this approach better let us now calculate the total number of factors (up to and including 10)

Number of factors for 1   = 0;
Number of factors for 2   = 0;
Number of factors for 3   = 0:
Number of factors for 4   = 2:
Number of factors for 5   = 0:
Number of factors for 6   = 3:
Number of factors for 7   = 0:
Number of factors for 8   = 3:
Number of factors for 9   = 2:
Number of factors for 10 = 3.

Therefore the total number of factors to 10 = 13.

Now these numbers are expressed with respect to linear notions of cardinal numbers as independent. However the trivial zeros properly relate to circular measurements with respect to ordinal numbers as interdependent with each other.

Therefore to obtain the appropriate "circular" conversion we multiply by 2π.

Therefore 13 now represents our estimate of the number of trivial zeros up to 62.83 (approx) on the imaginary scale (which is an indirect linear means of representing circular notions).

Using Andrew Odlyzko's tables, it can be easily shown that the actual number of zeros to 62.83... is in fact 14!

So our manual estimate (using the combined sum of factors up to 10) is surprisingly accurate.

And this is by no means an accident. I continued the calculations with the combined sum of factors (in groups of 10) up to 100. So I will now present in the first column the cardinal numbers in involved, in the second the corresponding number on the imaginary number scale to which they correspond (with respect to the trivial zeros). In the third I present the cumulative sum of factors (up to the cardinal number in question) serving as the estimate of trivial zeros. In the 4th I present the actual estimate and in the fifth the estimate from the formula for calculation of non-trivial zeros i.e. t/2π{ln (t/2π) – 1}. Finally in the last I present the deviation of my manual estimate from the actual number of zeros that occur.

No. to
Im. scale
(1) Or. zero est.
(2) Actual zeros
(3) Formula estimate
Deviation of (1) from (2)  
  ( – 1)
  ( – 3)
  ( – 2)
  ( – 3)
  ( – 5)
  ( – 3)
  ( – 5)
  ( – 5)
  ( – 5)
  ( – 4)

As can be seen from above the manual approach is surprisingly accurate (with this accuracy increasing in relative terms). As we have seen before, the formula is in fact stunningly accurate with the estimate generally being within 1 of the actual result.

This is why I have suggested before a slightly amended version of this formula i.e. t/2π{ln (t/2π) – 1} + 1.

I then carried out further estimates up to 200 (1256.63 on imaginary scale) with  the manual estimate of zeros up to this number 852 (with the actual result 861).

Interestingly the manual estimate - though relatively very close - is always less than the actual result.
In the next blog entry I will suggest a fascinating explanation of this small discrepancy from the actual result.

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