Complementary Formula for Zeta 2 Zeros (2)

Yesterday I indicated that an alternative ordinal (Type 2) approach exists for the estimation of primes up to a given number.

Again we saw that the well known formula for calculation of the Riemann (Zeta 1) zeros up to t on an imaginary scale is,

t/2π(log t/2π – 1),

This represents the "circular" holistic version of the corresponding simple analytic formula
n(log n – 1) for calculation of the combined frequency of natural factors up to n on a real scale (where n = t/2π).

We have a similar complementarity in evidence with respect to the (unrecognised) Zeta 2 zeros.

So (2t/π)/(log 2t/π – 1) represents the circular holistic formula for the corresponding frequency of the "converted" Zeta 2 zeros which can be used to calculate the frequency of prime numbers up to 2t/π on a real scale.

This can then be used as an alternative to the better known cardinal approach for calculating primes up to n on the same real scale i.e. n/log n – 1, where n = 2t/π.

Perhaps it would help to clarify the rationale with a practical example.

In earlier work, on the Zeta 2 zeros, I had manually calculated all roots (cos and sin part parts) up to p = 127.

So armed with this information I then set about calculating the prime numbered roots up to 127 with a view to calculating the sum of both sin and cos parts as an alternative means of measuring the frequency of primes.

The sum of the "converted" values of the cos part for all 127 roots of 1 = 80.85 (correct to 2 decimal places).

The sum of the "converted" values of the cos part for all prime numbered roots (up to 127) = 21 .55.

Therefore we can use these 2 measurements to estimate the frequency of prime numbers up to 81 to nearest integer) with both measurements rounded to nearest integer.as 22.

In fact in this case 22 is the exact number of primes up to 81!

I then calculated the corresponding sum for the "converted" values of all roots to 127 for the sin part, and then the sum of the prime numbered roots up to 127 for the sin part.

The sum of all "converted" natural number roots to 127 again = 80.25 (correct to 2 decimal places) with the sum of the prime numbered roots = 19.27.

This would suggest 19 prime numbers to 81 (which is 3 less than actual total).

However as the value of t increases, the sums for both cos and sin parts (with respect to natural and prime numbered "converted" roots) would converge closer and closer together so that  both estimates for prime frequency would be increasingly similar.


So once again, the key point about this exercise is that we have two complementary ways of calculating prime number frequency.

We have the standard cardinal (Type 1 approach) where we attempt to measure the number of primes up to a given number on the real scale.

However equally we have an (unrecognised) ordinal (Type 2) approach where we attempt to measure the "converted" frequency of the sum of prime numbered roots (for cos and sin parts) in relation to the overall sum of the natural numbered roots.

Now it is in relation to this 2nd approach that the formula,

 (2t/π)/(log 2t/π – 1) can be used.

So in relation to our example, t = 127.

Therefore this formula will measure the "ordinal" frequency of primes up to n = 2t/π on a "converted" real scale.

So 2t/π = 254/π = 80.85.

This confirms that the assumption that sum of both cos and sin parts for all natural numbered roots up to t = 127 approximates 2t/π is in fact already extremely accurate!

Our estimate for primes up to 81 from the formula = 24 to nearest integer (which represents an overestimate of 2 in this case).


So once again we have a linear (analytic) and holistic (circular) manner for calculating both the frequency of natural factors (to a given number) and the corresponding frequency of primes.

The linear (analytic) manner of estimating the combined frequency of natural factors up to n

= n(log n – 1).

The corresponding circular (holistic) manner fro calculating the combined frequency of natural factors up to t on an imaginary scale (using in fact the Riemann Zeta 1 zeros) is given as,

t/2π(log t/2π – 1) where n = t/2π.

Fascinatingly, we use an imaginary scale here, as the numerical measurement of factors (relating to dimensional transformation) is qualitatively distinct from number quantities (measured on a 1-dimensional linear scale).

So in fact we have established that a direct relationship exists as between the Riemann (zeta 1) zeros and the natural factors of the composite numbers.


We have also both linear (analytic) and holistic (circular) ways of calculating the corresponding frequency of primes.

The linear (analytic) manner for estimating the cardinal frequency of primes up to n,

= n/log n  – 1).

The corresponding circular (holistic) manner of estimating the ordinal frequency of primes up to t (where n = 2t/π),

= (2t/π)/(log 2t/π – 1).

So again we have established that a direct relationship exists as between the Zeta 2 zeros and the primes.


This however implies that a complementary relationship exists as between the Riemann (Zeta 1) zeros and the primes.

Equally it implies that a complementary relationship exists as between the Zeta 2 zeros and the natural factors (of the composite numbers).