Unexpected Link

In yesterday's blog entry, I showed a close and fascinating link as between the accumulated sum of the Riemann (Zeta 1) zeros and a corresponding sum relating to the aggregate of the factors of the composite nos.

I also demonstrated how the simple formula n(n + 1)(log n – 1)/2 can be used to estimate both sums.

Having completed this entry, I then considered another aggregate sum entailing the Riemann zeros.

Here we multiply each prime in ascending sequence by the matching entry from the list of (non-trivial) zeros (likewise arranged in ascending sequence).

So the 1st prime number is multiplied by the 1st Riemann zero, the 2nd prime by the 2nd Riemann zero, and so on.

Up to n in this case relates to the value of the primes in question.

So to illustrate this new aggregate up to 10, we find the sum of (2 * 14.13) + (3 * 21.02) + (5 * 25.01) + (7 * 30.42) with zeros expressed correct to 2 decimal places = 429.31.

Once again, because the non-trivial zeros are estimated with respect to the imaginary scale, t (where n = t/2π), to convert to n, we divide 429.31 by 2π = 68.33 (or 68 rounded to nearest integer).

In the table below I show these accumulated totals. (in col 2).

Up to n	Agg. of primes * by zeros = (2)	Accumulated sum of zeros = (3)	(2)/(2/π)   = (4)	(4)/(3) as %
10	68	91	107	117.58
20	228	493	359	72.82
30	634	1234	996	80.71
40	1228	2677	1928	72.02
50	2518	4221	3956	93.72
60	3736	6370	5869	92.14
70	5244	8767	8236	93.94
80	8079	12200	12691	104.02
90	10437	15858	16394	103.38
100	11808	20133	18548	92.13
110	15060	24958	23656	94.78

Then in col. 3 I show again the total accumulated sum of zeros (appropriately rescaled to n) as dealt with in yesterday's entry.

In fact, there is an unexpected link as between the two aggregates (in cols 2 and 3 respectively).

If we divide the  total in col 2 by 2/π (or alternatively multiply by π /2), we obtain a new set of figures (in col. 4) which bears close comparison with the previous aggregates measure for Riemann zeros (in yesterday's entry).

The link between the two sets of figures (2/π) is not accidental. We have already seen how this shows up as  very important measurement with respect to the Zeta 2 zeros (with the average reduced value for both the cos and sin parts of all the roots of 1 converging to 2/π).

In the final column (col 5), I show the relationship as % as between the measurements in cols. 3 and 4 respectively.

The formula for the new estimate in col. 2 (i.e. aggregate of each prime multiplied by corresponding non-trivial zero), is given as {n(n + 1)(log n – 1)}/π.