Simple Estimate of Frequency of Riemann Zeros (2)

In yesterday's blog entry, I demonstrated how a simple procedure based on the accumulated sum of the factors of composite numbers can be used to obtain a surprisingly accurate estimate of the frequency of the Riemann (non-trivial) zeros up to a given number (on the imaginary scale).

However one striking aspect of this procedure is that it always generates a number that is slightly less than the true actual occurrence of such zeros.

Now the very insight that led to my consideration of the use of such factors for estimation of the Riemann (Zeta 1) zeros sprung directly from a holistic approach.
Though in direct terms, this holistic approach represents the qualitative aspect of mathematical understanding, indirectly it can be effectively used, as in the case here of estimating the frequency Riemann zeros, in a surprisingly creative yet very simple quantitative manner.

When I was dealing with the corresponding Zeta 2 zeros, I was at pains to show how a strong complementary relationship existed as between numbers (representing dimensions) in both qualitative and quantitative terms.

So the true dimensional meaning of 3 for example is the qualitative ordinal notion of it thereby containing uniquely distinct 1st, 2nd and 3rd members. This is in direct contrast to the cardinal notion of 3 representing a whole quantitative meaning (where its individual units i.e. 1 + 1 + 1, thereby lack any qualitative distinction).

However the dimensional notion of 3 is then intimately connected with the 3 roots of 1 in an indirect quantitative manner. And in terms of the Type aspect 2 of the number system, these roots are represented as 1/3, 2/3 and 3/3 (i.e. 1^1/3, 1^2/3 and  1^3/3 respectively). So for example with respect to the 3rd dimension, a reciprocal connection exists as between its qualitative and quantitative expressions. 

This led me to again to the insight that in the corresponding case of the Zeta 1 zeros, that an important complementary role would likewise exist for the factors of the composite numbers.

So in a manner somewhat similar to the procedure outlined yesterday, I manually estimated a new accumulated sum for these factors.

For example whereas again yesterday in the case of  6, we have 3 factors in this new approach we simply divide this by the number to which these factors relate. Therefore the fractional result for this number therefore = 3/6 = 1/2.

To illustrate this approach better let us now calculate the relevant fractions of each number (up to and including 10)

Fraction for 1   = 0;
Fraction for 2   = 0;
Fraction for 3   = 0:
Fraction for 4   = 1/2:
Fraction for 5   = 0:
Fraction for 6   = 1/2:
Fraction for 7   = 0:
Fraction for 8   = 3/8:
Fraction for 9   = 2/9:
Fraction for 10 = 3/10.

Now when we accumulate the sum of fractions up to 10 we obtain 1.8972...

I the continued in this manner accumulating the sums of  these fractions up to 100.

The relevant results (in groups of 10's) for accumulated sums of fractions is listed below (with totals rounded to nearest integer in brackets).

Up to 10       1.8972...    (2)
      to 20       3.5059...    (4)
      to 30       4.7952...    (5)
      to 40       5.7694...    (6)
      to 50       6.5543...    (7)
      to 60       7.3061...    (7)
      to 70       7.8969...    (8)
      to 80       8.4638...    (8)  
      to 90       8.9872...    (9)
      to 100     9.4663...    (9)

Now one might question the significance of these totals!

However when appropriately interpreted they have an important role to play in the more precise prediction of the frequency of Riemann zeros.

As we have seen our initial simple estimate of these zeros (to any number), though predicting very accurately always give a total less than the true actual frequency.

Now when these corresponding cumulative sums sums of fractions are added in to the original totals, the result will always be a sum that is now slightly greater than the actual frequency.

Put another way the original sum and new sum (including fractional totals) sets a precise limit in which the true actual frequency value of  the Riemann zeros will occur.

Indeed casual inspection would suggest - even though local conditions can vary considerable - that on average the true actual total will fall half way as between the original estimate and the new larger estimate (that includes the fractional total.

So once again (using rounded values) I will present the position up to 100 representing original and  new totals (for comparison with actual frequency) and finally the new modified estimate based on the midpoint of both the original and new estimates
  

No. to	Im. scale	Or. zero est.	Actual zeros		New estimate	Mod. estimate
10	62.83…	13	14	15		14     (0)
20	125.66…	38	41	42		40 ( – 1)
30	188.49…	71	73	76		73     (0)
40	251.32…	106	109	112		109     (0)
50	314.15…	142	147	149		145 ( – 2)
60	376.99…	184	187	191		188 ( + 1)
70	439.82…	223	228	231		227 ( – 1)
80	502.65…	267	272	275		271 ( – 1)
90	565.48…	311	316	320		315 ( – 1)
100	628.31…	357	361	366		362 ( + 1)

As can be see from these figures, the actual frequency falls here between the original estimate (based on prime factors) and the new estimate (that includes fractional values based on original estimates).

Then by taking the midpoint of the original  and new estimates (i.e. the modified estimate) we get - what in fact is - an extremely accurate estimate of the actual frequency of these zeros. (The deviation from the actual occurrence of zeros is given in brackets).

Indeed in many cases, we get exactly the same estimate. In the other cases the difference is just 1 or at most 2 (which to a degree represents inevitable rounding errors).