## Thursday, May 15, 2014

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.