However this does not imply that the individual zeros of
these functions can be precisely calculated with reference to the individual
zeros of the Riemann zeta function.
One must remember that a complementary nature connects
the nature of the primes and the Riemann zeta zeros.
So each prime is characterised by the maximum amount of
individual uniqueness as it were consistent with the ordered nature of the
natural numbers.
Therefore though it is indeed possible to predict with a progressively
greater relative degree of accuracy the frequency of primes up to a given
number, in absolute terms the deviation as between the actual and predicted
number of primes tends to increase.
However by contrast each zeta zero is characterised by the
maximum amount of ordered sociability (between the primes and natural numbers)
consistent with each zero maintaining its individual uniqueness at an
individual level.
Therefore it is possible to predict the number of zeta zeros
up to a given magnitude not only accurately in relative terms but likewise also
in an absolute manner.
Thus the very nature of the zeta zeros is to keep
reconciling the individual uniqueness of the zeros at a local with the maximum
amount of order possible at a collective level.
Thus deviations at a local level keep getting cancelled out
in terms of maintaining this order at the collective level.
However this entails that though we can calculate almost
exactly in absolute terms the frequency of Riemann zeros and the zeros of other
Dirichlet functions up to any given level, because of the uniqueness of each zero, this does not apply so well at the
individual level.
However even here, from knowledge of the individual zeta
zeros, we can however, through knowledge of the
conductor, make a reasonable attempt to approximate the individual zeros of the
other Dirichlet functions.
For example to illustrate I made an estimate of the first 10
zeros of the Dirchlet Beta Function.
As we have seen the Dirichlet Beta Function as a conductor
of 4. So starting with the 3rd Riemann zero I estimated each 4th
zero up to the 39th zero (giving 10 in all).
Then I divided each these zeros by 4 to provide an estimate
of the corresponding zero of the Dirichlet Beta Function
Ordinal number of zeta zero
|
Zeta zero (correct to 3 decimal places)
|
Zeta zero/4
|
First 10 Dirichlet Beta Function zeros
|
3
|
25.011
|
6.253
|
6.001
|
7
|
40.919
|
10.230
|
10.264
|
11
|
52.970
|
13.243
|
13.002
|
15
|
65.113
|
16.278
|
16.353
|
19
|
75.705
|
18.926
|
18.285
|
23
|
84.735
|
21.184
|
21.445
|
27
|
94.651
|
23.663
|
23.274
|
31
|
103.726
|
25.932
|
25.726
|
35
|
111.875
|
27.969
|
28.358
|
39
|
121.370
|
30.343
|
29.662
|
The estimates here, though not exact are however quite good with some local deviation however in evidence.
However there is a certain degree of arbitrariness in the manner that I started
with the 3rd Riemann zero in obtaining my estimates.
Perhaps the most unbiased approach is to obtain a mean or median
within the range of conductor values.
So in this case this would imply getting an average of each successive
group of 4 Riemann zeta values (before then dividing by 4) or alternatively the
average of the 2nd and 3rd values within each group of
before dividing by 4. Though the estimates would not be quite as good as above they would still be reasonably accurate. And this approach could then be generalised to approximate the individual zeros for any Dirichlet L-function.
No comments:
Post a Comment