Return to Dirichlet Beta Function

Let us return once more to the Dirichlet Beta function i.e.

L(χ,s)

= 1 − 3-s + 5-s − 7-s + 9-s − 11-s + 13-s − 15-s + 17-s − 19-s + 21-s − 23-s + 25-s − 27-s + ...

Only the odd natural numbers are involved here alternating as between positive and negative values.

So we have two distinct series i.e. 1, 5, 9, 13, 17, … and 3, 7, 11, 15, 19, … etc.

All of the odd numbers represent ½ of the natural numbers; then each of these two series of odd numbers represent ¼ of the natural numbers.

We can now attempt to estimate the frequency of factors of both of these series in two ways.

1. We can do so with respect to the cumulative total of factors up to n. The formula that can be used here to approximate the no. of factors involved = n(ln n − 1) 

2. We can alternatively use the cumulative total of zeta zeros up to t (where n = t/2π). The formula that can be used here to give an extremely accurate answer for  the frequency of zeros =   t/2π(ln t/2π − 1).  

In the following illustration I manually counted the cumulative frequency of factors (excluding 1) of the 1^st series 1, 5, 9, 13, 17, 21, … in blocks of 50 up to n = 300.

I then manually counted the cumulative frequency of all the natural nos. (except 1) again in blocks of 12.5 up to n/4 = 75.

This latter count of all natural number factors to n/4 then provided an estimate of the corresponding count of the odd factors in this first series.

For example there are 29 odd factors (belonging to the 1^st series) up to n = 50 and the corresponding count of natural no. factors (except 1) to n/4 = 12.5 is 23. So this computation involving all natural no. factors provides an estimate of the frequency of factors 5, 9, 13, 17, 21, …

We could also use the frequency of zeta zeros up to t/4 as an estimate.

As t = 2nπ, t/4 = nπ/2

So again to measure the frequency of factors 5, 9, 13, 17, 21, … up to 50 we calculate t/8π(ln t/8π − 1). In the first instance this serves as an accurate measurement of the frequency of zeros up to t/4 which then likewise serves as an approximation of the frequency of the odd factors in the series to n. And there are 19 zeta zeros to t/4 (i.e. 50π/2 = 25π) serving as a not so accurate estimate in this case. However as n increases the approximation steadily improves.

n	Total no. of factors 5, 9, 13,17, 21, …to n	Estimated total using natural no. factors to n/4		Zeta zero estimate to t/4
50	29	23	19
100	79	62	55
150	132	106	98
200	189	157	146
250	247	207	196
300	308	264	249

Now it is interesting to note that this 1^st series of odd numbers follows the distribution of the Riemann zeta zeros.

The second task is then to estimate the frequency of odd factors in the 2^nd series i.e. 3, 7, 11, 15, 19, …

Again I measured this in blocks of 50 up to n = 300. I first calculated the actual no. of factors (in the 2^nd series) up to n. I then used an estimate that is directly based on the Dirichlet distribution.

In this case I calculated all the even factors - not to n/4 - but rather to n, before then dividing the total by 4 to get an estimate for the actual no. of factors. This estimate turns out to be remarkably accurate.

For example the actual frequency of the odd factors 3, 7, 11, 15, 19, … to 50 = 39.

And the corresponding estimate using all factors (except 1) to 50 is also 39. And this is no coincidence as the table below indicates.

Also the estimate using the Dirichlet Beta zeros as an estimate is also very accurate in this case where we count all the zeros to t (where t = 2nπ) before dividing the total by the conductor of the function (i.e. N = 4).

n	Total no. of factors 3, 7, 11,15, 19, …to n	(Estimated total using natural no. factors to n)/4		Zeta zero estimate to t before division by 4
50	39	39	36
100	95	96	90
150	155	155	150
200	222	222	215
250	287	286	283
300	368	364	353

One further interesting fact is that the sum of all Riemann zeta zeros to 2t minus the sum of all zeta zeros to t = the sum of all Dirichlet Beta zeros to t.

For example there are 79 zeta zeros to 200 and 29 zeta zeros to 100. And the difference = 50 is the frequency of Dirichlet Beta zeros to 100.