Estimating Sum of Riemann Zeros (2)

In my last blog entry, I mentioned how the simple formula n(n + 1)(log n – 1)/2 can be used to estimate two aggregate sums with respect to both the factors of composite numbers and the sum of the Riemann (Zeta 1) zeros.

I illustrated this approach with respect to the aggregate sum of factors showing how it is compiled for the composite numbers up to 10.

Once again 4 has 2, 6 has 3, 8 has 3, 9 has 2 and 10 has 3 factors. So we obtain therefore the sum  (4 * 2) + (6 * 3) + (8 * 3) + (9 * 2) + (10 * 3) = 8 + 18 + 24 + 18 + 30 = 98.

I will now likewise illustrate up to n = 10, with respect to the corresponding aggregate of the Riemann zeros.

We have to remember that these zeros are measured on the imaginary scale up to t where n = t/2π. Thus we add the zeros up to t = 62.83, i.e. 14.13 + 20.02 + 25.01 + 30.42 + 32.94 + 37.59 + 40.92 + 43.33 + 48.01 + 49.77 + 52.97 + 56.45 + 59.35 + 60.83 (correct to 2 decimal places) = 571.74.

Then to express this result with respect to n, we divide by 2π to obtain 90.96 = 91 (to nearest unit).
    

Up to n	Acc. sum of factors (1)	Acc. sum of Riemann zeros (2)	Formula Est. (3)	(2)/(1)     as %	(3)/(2) as %
10	98	91	72	92.86	79.12
20	499	493	419	98.80	84.99
30	1355	1234	1117	91.07	90.52
40	2620	2677	2205	102.17	82.37
50	4277	4221	3713	98.69	87.96
60	6459	6370	5663	98.62	88.90
70	9038	8767	8073	97.00	92.08
80	12073	12200	10956	101.05	89.80
90	15947	15858	14322	99.44	90.31
100	20367	20133	18206	98.85	90.43
110	24608	24958	22591	101.42	90.52

In the above table, I show the results (in 10's) up to n = 110, for both the aggregate sum of  factors of the composite nos. (col. 2) and the corresponding sum of Riemann zeros (col. 3) with values rescaled to n.

Then in col. 4, I show the estimated aggregate sum using the formula, i.e.{n(n + 1)(log n – 1)}/2.

Then in col 5, I show the % accuracy of the actual sum of factor values and the corresponding sum of zeros.

These values compare vary well indeed. Sometimes the factor value sum exceeds that of the sum of zeros (and likewise the sum of zeros may exceed the sum of factor values). However even though we are still at a very early point on the n scale, the two sets of values already have converged very close to each other (in the region of 99% accuracy) as indicated in col. 5..

The formula estimate is not quite so accurate (with respect to both factor sums and zero aggregate sums). However it gives a consistent estimate (with % accuracy gradually improving) that already - as can be seen from the final column (col. 6) - is about 90% accurate.

Of course - as was the case for so long with the prime number theorem - I have not provided actual proofs of what is indicated by the data.

However I would confidently assert that the actual sums (with respect to both the factors and zero sum aggregates) would eventually approach 100% accuracy (in relative terms).

Likewise with only slightly less confidence, I would expect that the formula estimate (at high n) would also approach 100% accuracy (in relative terms) with respect to the prediction of both aggregates.