Once again this formula can be given as:

t/2π{(log t/2π) – 1}

I then suggested in a previous blog entry that perhaps a slightly more accurate version can be given through the addition of 1,

i.e. t/2π{(log t/2π) – 1} + 1.

Thus using this (amended) version the number of non-trivial zeros up to 100 i.e. where t = 100 on the imaginary line,

= 15.9154943...(2.7672931... – 1) + 1 = 29.127...

The actual number of zeros up to 100 = 29! so we can see in this case how the formula gives a surprisingly accurate answer.

And as I conveyed in "Stunning Accuracy" this accuracy remains even at the highest values of t for which non-trivial zeros have been yet calculated.

However though this formula has been explicitly formulated with respect to the Zeta 1 Function, once again strong complementary links can be shown to exist with respect to the Zeta 2 Function,

i.e. ζ

_{2}(s) = 1 + s

^{1 }+ s

^{2 }+

^{ }s

^{3 }+…… = 1/(1 – s).

As we have seen in a qualified sense. ζ

_{2}(s) = 0 for non-trivial roots of the equation (where the number of roots of 1 is prime and we consider all t – 1 roots except 1).

In this case the t – 1 non-trivial roots correspond to the finite equation,

1 + s

^{1 }+ s^{2 }+^{ }s^{3 }+.... +^{ }s^{t }^{– 1 }= 0.
Then when we continue with regular cycles of t terms, these t – 1 roots will likewise correspond to the infinite equation,

1 + s

^{1 }+ s^{2 }+^{ }s^{3 }+…… = 0.
In yesterday's blog entry, we saw how the number of possible values generated = t * (t – 1)

For example where t = 3, the two non-trivial roots correspond to the finite equation,

1 + s

^{1 }+ s^{2 }= 0.
So 3 * 2 possible values (= 6) can be generated in this case.

Firstly, we obtain the 3 values from substituting the root (– .5 + .866i ) as a value for s,

to obtain 1, – .5 + .866i and – .5 – .866i respectively.

Then we substitute the root (– .5 – .866i ) as a value for s,

to obtain 1, – .5 – .866i and – .5 +.866i respectively.

The finite Zeta 2 Function is expressed in terms of natural number powers of s (from 1 to t – 1).

However strictly the non-trivial zeros relate to values of t that are prime!

Now the average spread (or gap) between primes in the region of t = log t.

So when we calculate the value of t terms with respect to the log t roots of t, we get

= t * {log t – 1}

However, as we have seen the roots of 1 are defined with respect to the unit circle with circumference 2π.

Therefore to convert from circular to linear units of measurement we divide t by 2π!

So therefore, when expressed in linear terms, the number of possible values generated up to s

^{t }^{– 1 }for frequency of prime (i.e. log t) roots
= t/2π{(log t/2π) – 1}

and this is identical with the formula for frequency of non-trivial zeros with respect to the Zeta 1 Function.

This provides the holistic means of understanding what the non-trivial zeros with respect to the Zeta 1 Function truly represent. In fact, from the dynamic interactive perspective, through which they are appropriately viewed, they represent the opposite (complementary) aspect of prime numbers, or - as I have expressed before - the perfect shadow counterpart number system to the primes.

We start by viewing each prime in linear terms as a specific locally defined independent number (i.e. with no factors other that itself and 1) which thereby serves as a unique building block for the (composite) natural numbers.

However here at the other extreme, we have a set of counterpart numbers in circular terms that possess - by contrast - a collective holistically defined identity as a set of numbers, that serve to perfectly reconcile the primes and natural numbers as fully interdependent with each other.

Now because the Zeta 1 zeros are indirectly expressed in a linear (imaginary) format, it is difficult from this perspective to appreciate their true holistic identity.

However it will help to recognise that this linear imaginary identity (in Zeta 1 terms) represents just an indirect way of translating their inherent circular holistic identity (in Zeta 2 terms).

As we know the frequency of the primes lessens as we ascend the (real) number line. In the region of 100 we would expect an average gap of less than 5 as between successive primes; then in the region of 1,000 it would be 7 (approx.) and then in the region of 1,000,000, 14 (approx.)

Now this average gap as between primes in the region of t is given as log t.

However, if for example in the region of 1,000, we expect an average gap of about 7 between each successive prime (as an independent number with no factors), then this equally entails 6 other composite numbers (expressing the interdependence of a unique set of prime factors).

As we have seen, whereas the independent aspect of number is expressed through the Type 1, the interdependent aspect is expressed through the Type 2 approach. And in the context of the Zeta Function these two aspects relate to the Zeta 1 and Zeta 2 Functions respectively.

Thus the interdependence attaching to the composite numbers is appropriately expressed through the Zeta 2 Function.

Thus in the region of 1,000 we would express the interdependence of the 6 composite numbers by the finite equation which provides in the case of 7, the 6 non-trivial roots of 1,

1 + s

now with t = 1,000 in this case the 6 terms in s are given as log t – 1 .

Now there are seven distinct values in the equation and we can express this with respect to each of the 6 possible values (as roots) of s. thus in total we have

log t * (log t – 1) values.

However if we now continue up t0 t (in circular groups of 7) the number of possible values =

t * (log t – 1).

However as these values are based on the circle of unit radius, to convert to linear terms we divide t by 2π.

In this format the number of possible values is

t/2π{(log t/2π) – 1}, which is the well-known formula for calculation of the frequency of Zeta 1 zeros.

Thus we can perhaps see here that the Zeta 1 zeros serve as an indirect quantitative representation (in a necessarily imaginary linear format) of the collective holistic nature of the number system (which is directly qualitative in nature).

So therefore though each individual Zeta 1 zero is quantitative in nature (of an imaginary nature), the true significance of these zeros is in their overall collective holistic nature (which is qualitative).

And the direct way of appreciating this qualitative aspect is through the Zeta 2 Function!

We start by viewing each prime in linear terms as a specific locally defined independent number (i.e. with no factors other that itself and 1) which thereby serves as a unique building block for the (composite) natural numbers.

However here at the other extreme, we have a set of counterpart numbers in circular terms that possess - by contrast - a collective holistically defined identity as a set of numbers, that serve to perfectly reconcile the primes and natural numbers as fully interdependent with each other.

Now because the Zeta 1 zeros are indirectly expressed in a linear (imaginary) format, it is difficult from this perspective to appreciate their true holistic identity.

However it will help to recognise that this linear imaginary identity (in Zeta 1 terms) represents just an indirect way of translating their inherent circular holistic identity (in Zeta 2 terms).

As we know the frequency of the primes lessens as we ascend the (real) number line. In the region of 100 we would expect an average gap of less than 5 as between successive primes; then in the region of 1,000 it would be 7 (approx.) and then in the region of 1,000,000, 14 (approx.)

Now this average gap as between primes in the region of t is given as log t.

However, if for example in the region of 1,000, we expect an average gap of about 7 between each successive prime (as an independent number with no factors), then this equally entails 6 other composite numbers (expressing the interdependence of a unique set of prime factors).

As we have seen, whereas the independent aspect of number is expressed through the Type 1, the interdependent aspect is expressed through the Type 2 approach. And in the context of the Zeta Function these two aspects relate to the Zeta 1 and Zeta 2 Functions respectively.

Thus the interdependence attaching to the composite numbers is appropriately expressed through the Zeta 2 Function.

Thus in the region of 1,000 we would express the interdependence of the 6 composite numbers by the finite equation which provides in the case of 7, the 6 non-trivial roots of 1,

1 + s

^{1 }+ s^{2 }+^{ }s^{3}+ s^{4 }+ s^{5 }+^{ }s^{6 }= 0now with t = 1,000 in this case the 6 terms in s are given as log t – 1 .

Now there are seven distinct values in the equation and we can express this with respect to each of the 6 possible values (as roots) of s. thus in total we have

log t * (log t – 1) values.

However if we now continue up t0 t (in circular groups of 7) the number of possible values =

t * (log t – 1).

However as these values are based on the circle of unit radius, to convert to linear terms we divide t by 2π.

In this format the number of possible values is

t/2π{(log t/2π) – 1}, which is the well-known formula for calculation of the frequency of Zeta 1 zeros.

Thus we can perhaps see here that the Zeta 1 zeros serve as an indirect quantitative representation (in a necessarily imaginary linear format) of the collective holistic nature of the number system (which is directly qualitative in nature).

So therefore though each individual Zeta 1 zero is quantitative in nature (of an imaginary nature), the true significance of these zeros is in their overall collective holistic nature (which is qualitative).

And the direct way of appreciating this qualitative aspect is through the Zeta 2 Function!

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