This frequency as we have seen approximates closely the
corresponding frequency of all factors (odd and even) to n/2.
So for example, if n = 300, the accumulated frequency of all
the odd factors of the natural numbers up to 300, approximates closely the
corresponding frequency of all factors (odd and even) to 150.
Is implies therefore that to approximate the accumulated
frequency of the even factors to 300, we can thereby count the frequency of all
factors from 151 to 300.
Now the accumulated frequency of all odd factors to 300 is
680 (excluding as in all cases the trivial case where 1 is a factor). And if we
count all factors (odd and even to 150) we get 629. So the relative accuracy of
our result here is 92.5%. So the all factor underestimates the odd factor total
in this case. But the approximation steadily increases towards 100% (in
relative terms) as we progressively increase the value of n.
Likewise the accumulated frequency of all even factors to
300 is 803. However if we now count the frequency of all factors (odd and even)
from 151 to 300 we get 854.
So the relative accuracy of our result is here 94% with the
all factor now overestimating the frequency of even factors. However, once
again the approximation steadily increases towards 100% as we progressively
increase the value of n.
Alternatively we could say that if t = 2πn the frequency of
zeros to t/2 = πn approximates the corresponding accumulated frequency of all
odd number factors to n.
Also the frequency of zeros t/2 to t approximates the
corresponding frequency of all even number factors to n.
Illustrating further, as we saw in the last entry, the
accumulated frequency of factors up to n (not containing 3 or multiples of 3)
can be measured by counting all factors to 2n/3.
Therefore again where n = 300, the accumulated frequency of
factors (not containing 3 or multiples of 3) to 300 can be approximated by
counting all factors up to 200.
This likewise implies that the accumulated frequency of
those factors (comprising 3 or multiples of 3) can be approximated by counting
all factors from 201 to 300.
The cumulative frequency of factors (other than those
containing 3 or multiples of 3) to 300 = 1001. And the cumulative frequency of
all factors to 200 = 913.
So the relative accuracy of this latter estimate = 91.2% which
is an underestimate but gradually increases towards 100% (in relative terms) as
n increases.
And the corresponding cumulative frequency of factors (that
are 3 or multiples of 3) to 300 = 479. And the cumulative frequency of all
factors from 201 to 300 = 570.
So the relative accuracy of this latter estimate is 84%
which is an overestimate but which again will approach 100% (in relative terms)
as n progressively increases.
Alternatively we could say that if t = 2πn the frequency of
zeros to t/3 = (2π/3)n approximates the corresponding accumulated frequency of
all number factors (that do not include 3 as a divisor) to n.
Also the frequency of zeros 2t/3 to t approximates the
corresponding frequency of all number factors (that include 3 as a divisor) to
n.
Then we also saw in the last entry that if we exclude all
factors containing either 2 or 3 as factors that the cumulative frequency of
all factors to n/3 gives a good approximation.
Therefore again with n = 300 the cumulative frequency of all
factors to 100 provides an estimate of those factors to 300 (that exclude 2 or
3 as divisors).
The cumulative frequency of all factors (excluding 2 or 3 as
divisors) to 300 = 403.
And the corresponding cumulative frequency of all factors to
100 = 382.
Therefore the relative accuracy is here 94.8% and approaches
ever closer to 100% as n becomes progressively larger.
We can also approximate the cumulative frequency of all
those factors (with 2 or 3 as divisors) to 300 through the corresponding
calculation of all factors between 101 and 300.
Now the cumulative frequency of those factors (with 2 or 3
as divisors) to 300 = 1080.
And the cumulative frequency of all factors between 101 and
300 = 1101.
So the relative accuracy is here 98.1% and approaches ever
closer to 100% as n progressively increases.
Alternatively we could say that if t = 2πn the frequency of
zeros to 2t/3 = (4π/3)n approximates the corresponding accumulated frequency of
all number factors (that do not include 2 or 3 as a divisor) to n.
Also the frequency of zeros t/3 to t approximates the
corresponding frequency of all number factors (that include 2 or 3 as a
divisor) to n.
In principle these ideas can be extended to any combination
of factors.
For example say we want to approximate the accumulation of
all factors (excluding 3, 7 or 11 as divisors) to n
Now the exclusion of factors containing 3 = 1/3 of divisors.
Then 1/3 of numbers divisible by 7 are also divisible by 3
so fractions divisible by both 3 and 7 = 1/3 + 2/3(1/7) = 7/21 + 2/ 21 = 9/21 =
3/7.
Then the fraction of numbers divisible by 3, 7 or 11 = 3/7 +
1/11(4/7)
= 33/77 + 4/77 = 37/77.
Therefore the fraction of natural numbers not divisible by
3, 7 or 11 = 40/77.
So to approximate the frequency of factors to n (that do not
have 3, 7 or 11) as divisors, one accumulates all factors to (40/77)n.
Thus in the convenient case where n = 77, one would count
the factors of all numbers to 40 = 118.
And the actual number of such factors to 77 = 131. So the
relative accuracy is 90.1% Then to approximate those factors with 3, 7 or 11 as
divisors, one would count all factors from 41 to 77 = 153. And the actual number of such factors = 140.
So the relative accuracy = 93.6%, which would improve towards 100% as n becomes
progressively larger.
And as before one can use zeta zeros to approximate the
answer in both cases.
So in the former case we would
count the zeta zeros to 80π i.e. 251.3 = 109 with a relative accuracy = 83.2%.
Then in the latter case we would
count the zeros from 80π to 154π, i.e. from 251.3 to 483.8 i.e. 261 – 109
= 152. Therefore the relative accuracy here = 99.3%.
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