Estimating Individual Zeros of Dirichlet L-Functions

Yesterday I showed how the frequency of zeros of all Dirichlet L-Functions (degree 1) is intimately related through the conductor to the corresponding frequency of (non-trivial) zeros of the Riemann zeta function.

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 3^rd Riemann zero I estimated each 4^th 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 3^rd 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 2^nd and 3^rd 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.