Calculating Frequency of Zeros for Dirichlet L-Functions

Some time ago I was interested to find computations of the Zeros for the Dirichlet Beta Function by Anthony Lander (the first 320 zeros).

This table of zeros enabled me to establish a close connection with the better known (non-trivial) zeros of the Riemann Zeta Function.

Now again the Dirichlet Beta function is given as

L(χ, s) = 1/1^s – 1/3^s + 1/5^s – 1/7^s + …   = 1/(1 + 1/3^s) *  1/(1 – 1/5^s) * 1/(1 + 1/7^s) * …, where χ is the Dirichlet character and s the power associated with each term

Now using the established terminology this represents a Dirichlet L-function of degree 1.

So when on dividing each successive number n by 4, if number is even then χ(n) = 0; if remainder is 1, χ(n) = 1; if remainder is 3, χ(n) = – 1.  

So the conductor N, which is the number by which n is divided, in this case is 4.

Now this conductor is directly relevant in terms of establishing the relationship of zeros in the Dirichlet Beta Function and zeros in the Riemann zeta function respectively.

Quite simply if the total frequency of zeros in the Riemann Zeta function up to n = k, the corresponding frequency of the zeros in the Dirichlet Beta function up to n/4 = k/4.

For example there are 649 zeros for the Riemann zeta function up to 1000.

So here n = 1000 and k = 649.

Therefore this entails that the corresponding number of zeros in the Dirichlet Beta

Function to 250 (i.e. n/4) = 649/4 = 162 (to the nearest integer).

And Lander’s table does indeed show exactly 162 Dirichlet Beta zeros up to 250.

Again there are 79 zeros for the Riemann zeta function up to 200.

This entails that the frequency of Dirichlet Beta zeros up to 50 = 79/4 = 20 (to the nearest integer).

And again Lander’s table shows exactly 20 such zeros to 50.

So there seems a pretty exact correspondence here as between the frequency of both sets of zeros.

In fact this correspondence seems to universally apply to all Dirichlet L-functions of degree 1.

So therefore in general terms once again if the frequency of zeta zeros up to n is k, the frequency of the alternative set of zeros up to n/N (where N is the conductor) is k/N.

So, L(χ, s) where χ is the Dirichlet character with label 7, 6 at the LMFDB  is given as,

1 + 1/2^s – 1/3^s + 1/4^s – 1/5^s – 1/6^s + 1/8^s + …

So the conductor here is 7.

Now frequency (k) of zeta zeros to n = 210 is 85.

Therefore the corresponding frequency of this alternative function with conductor 7 to 210/7 = 30 is 85/7 = 12 (to the nearest integer).

And in the list of the first few zeros of this function at the LMFDB, there are indeed exactly 12 zeros up to 30.

As I have said this seems a universal feature with respect to all Dirichlet L-functions of degree 1.

In fact this remains the case even where imaginary or complex numbers are involved.

For example the following is the Dirichlet L-function with label 5.3 from the LMFDB

L(χ,s)  = 1 − i·2^-s + i·3^-s − 4^-s + 6^-s − i·7^-s + i·8^-s − 9^-s + 11^-s − i·12^-s + i·13^-s − 14^-s + 16^-s − i·17^-s + i·18^-s − 19^-s + ⋯

So the conductor here is 5.

Therefore if n = 200, the frequency of zeros of this function to n/N i.e. 200/5 = 40 is given as k/N = 79/5 = 16 (correct to nearest integer).

And once again in the list of the first few zeros of this function there are exactly 16 up to 40.

And the following the Dirichlet L-function with label 7.5 from the LMFDB

L(χ,s) =

1 + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s − 6-s +  8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s −  13-s + 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯

So the conductor here is 7.

As we have seen there are 85 zeros to 210. Therefore once again we would expect up to 210/7 = 30, 85/7 = 12 zeros. And the table of first few zeros at LMFDB does indeed show exactly 12 zeros for this function up to 30.

As we have seen before the formula for calculating zeta zeros up to a given number is remarkably accurate (even in absolute terms)

So therefore if the frequency of zeros to t is given as  t/2π(ln t/2π − 1) then the corresponding frequency of zeros of an alternative Dirichlet function of degree 1 (with conductor N) 
= t/2πN((ln t/2π − 1).