Estimating Sum Of Riemann Zeros

In earlier blog entries, "Simple Estimate of Frequency of Riemann Zeros 1" and "Simple Estimate of Frequency of Riemann Zeros 2", I demonstrated how the frequency of the non-trivial (Zeta 1) zeros are closely related to the accumulated sum of the number of factors contained in the composite numbers.

I then considered the possibility of a simple formula which would estimate the sum of these zeros (up to a given number on the imaginary scale).
Also given the strong link as between the non-trivial zeros and the factors of the composite numbers, this formula would equally apply to an important aggregate with respect to the factors of composites.

As I have stated before the simple formula for estimating the accumulated frequency of factors of the composite numbers bears a complementary relationship with a similar type formula for the estimation of the frequency of primes (to a given number)

So n/(log n – 1) measures the frequency of primes (to n on the real scale)) whereas n(log n – 1) measures the corresponding (accumulated) frequency of the factors of the composite numbers (to n on the same scale).

Now to adjust this latter formula for the estimation of the non-trivial zeros up to the point t (on the imaginary scale) we set n = t/2π  

So the frequency of non-trivial zeros (up to t) is thereby given as,

 t/2π{log(t/2π) – 1}

Now I have suggested the addition of 1 to this formula giving,

 t/2π{log(t/2π) – 1} + 1 as the recommended version.

This gives stunningly accurate estimates of the frequency of these zeros - not only in relative - but also in absolute terms. Even at the highest values on the imaginary scale in the tables provided by Andrew Odlydzko  estimates using this formula are frequently exactly correct in absolute terms (and generally are accurate to within 1 of the correct value).

So unlike the primes which occur unpredictably, leading to a  discontinuous jump in the step function representing their actual frequency, the non-trivial zeros lie at the other extreme, with respect to the smoothing out of such irregular discrete behaviour. Not surprisingly therefore, the frequency of such zeros can then be very accurately predicted through a continuous function (such as the one here suggested).

However though the random zeros therefore represent an extreme in terms of the notion of continuous order, this can only be interpreted in a relative rather than absolute manner.
Thus we could equally draw, as with the primes a step function to exactly represent (in absolute terms) the frequency of the non-trivial zeros. Thus with the occurrence of each new non-trivial zero a discontinuous increase of 1 will take place. However because these zeros are located as well as is possible to ensure complete order with respect to behavior in the number system, we can thereby expect to estimate to an extraordinary degree of accuracy the actual occurrence of the non-trivial zeros.

Bearing this in mind, it is now possible to suggest the appropriate aggregate (with respect to the factors of the composite numbers) to which the sum of the non-trivial zeros should thereby correspond.

As we have seen the composite numbers (with factors) complement the primes (with no factors).

In this sense the composites veer towards the polar aspect of ordered behaviour (with respect to the number system) in contrast to the primes which veer towards the opposite pole of random behaviour.

However this ordered behaviour with respect to factors occurs in a somewhat discontinuous manner.

So we start with 1, 2 and 3 (with no true factors yet in evidence). Then we reach 4 (which immediately contains 2 factors). 5 is once again prime, and then we hit 6 (with 3 factors).

So the factors of the composite numbers always occur as immediate discontinuous multiple amounts (in striking contrast to the random behaviour of the primes).

So if we were to draw a step function for the composite numbers, we would encounter far more discontinuous steps on our journey. Furthermore, rather than all these steps being measured in single units (as with the primes) the heights of these steps vary considerably. So for example whereas 10 has a step of height 3) 12 has a step of height 5. In other words 10 has 3 factors (2, 5 and 10) whereas 12 has 5 factors (2,3,4,6 and 12).

So the considerable task with respect to the number system is the reconciliation of both the random aspect of the primes, representing their individual behaviour and the ordered aspect of the composites, representing their collective behaviour (through common factors).

Now this vitally important task is provided for the number system as a whole through the non-trivial (Zeta 1) zeros.

As I have repeatedly stated we can only appreciate this properly in a relative dynamic interactive manner.

So each non-trivial zero represents therefore a point where the notion of random (independent) behaviour with respect to the number system is fully reconciled  - as is finitely possible in a relative approximate manner - with the corresponding notion of ordered (interdependent) behaviour.

I have also repeatedly stated that these two aspects are properly quantitative and qualitative with respect to each other.

It is certainly true (within isolated frames of reference) that both the primes and non-trivial zeros can individually be given a quantitative identity.

However just like the two turns at a crossroads are left and right with respect to each other, from a dynamic experiential context both the primes and the non-trivial zeros are quantitative and qualitative (and qualitative and quantitative) with respect to each other.

So the quantitative (analytic) appreciation of this relationship comes from (initially) interpreting the primes and non-trivial zeros within isolated frames of reference. However the true qualitative (holistic) appreciation of their collective interdependence only can come from a dynamic approach (viewing both poles in a truly complementary manner).


Thus the whole mindset of Conventional Mathematics at present is sadly unsuited to the proper appreciation of the relationship of the primes and the non-trivial zeros.
In formal terms such Mathematics takes place within a linear (1-dimensional) framework. Here, merely isolated frames of reference interpreted analytically in a merely quantitative manner are considered.

However just as we can view the relationship of the primes and the non-trivial zeros for the number system as a whole (in Type 1 terms) equally we can do this within each prime.

Here each prime is considered as a group of related members in an ordinal natural number fashion.

So once again for example 3 as prime is composed thereby of  1st, 2nd and 3rd members.

We have here again the fascinating relationship of quantitative and qualitative notions. So we have 1st, 2nd and 3rd members (in ordinal terms) belonging to a group of 3 (in cardinal terms).

The Zeta 2 zeros here represent the corresponding roots of 1 (with the non-unique root of 1 excluded).

So the 3 roots 1, – .5 + .866i and – .5 – .866i , can each be given a relatively independent identity in quantitative terms.

However the collective identity of these roots is expressed through their sum = 0.

So this sum of roots strictly has no quantitative identity.

However once again (individual) independence and (collective) interdependence can only be appreciated in a relative, rather than absolute manner. So even in demonstrating the collective interdependence of the roots we must arbitrarily fix one position (i.e. the 1st) in an independent manner before considering subsequent interdependent relationships.

So coming back to our original task of estimating the sum of the Riemann (Zeta 1) zeros, the trivial zeros represent a  certain smoothing out with respect to the composite factors.

Therefore instead of two factors at the same number point (as with for example 4), we have two  points chosen at other unique locations.

So with respect to the real scale, corresponding to 2 points at 4, we have the first non-trivial zero corresponding to 2.249... (i.e. 14.134725/2π and  3.346... (i.e. 21.022040/2π).

Now again, the reason why these non-trivial zeros lie on an imaginary line, is because of their true holistic quality. And in holistic mathematical terms, the imaginary notion relates to the indirect attempt to represent such holistic meaning in a linear analytic manner.

So once again in direct terms, the Riemann zeros should be interpreted in a holistic (i.e. dynamically interactive) manner for their proper comprehension.

Therefore if we sum up all the non-trivial zeros (i.e. divided by 2π) to n on the real scale, this should correspond to the sum of each composite number multiplied by its corresponding number of factors (also up to n)

Therefore the sum of non-trivial zeros up to 10 * 2π on the imaginary scale (where the total is divided by 2π), should then correspond approximately to the sum of composite factors (each multiplied by its number of factors) up to 10,

i.e. (4 * 2) + (6 * 3) + (8 * 3) + (9 * 2) + (10 * 3) = 8 + 18 + 24 + 18 + 30  = 98.

The corresponding sum of trivial zeros (adjusted to the real scale) = 91. So this provides already a fairly good approximation.

The simple formula that I then suggest to approximate both measurements is:

{n(n + 1)(log n – 1)}/2.

We will return to further consideration of this formula in a future entry.