Tables of approximate values of the first zeros onthe critical line of some primitive Dirichlet L-series

Introduction

This page presents the results of my efforts to compute the first zeros, on the critical line, of some Dirichlet L-series. Only some L-series associated with Dirichlet characters [1] were considered. The Dirichlet L-series associated with the simplest character is the well-known zeta function. Riemann, in a path-breaking paper [2], conjectured that the non-trivial zeros of this function have real part equal to 1/2 (the critical line). This constitutes the famous Riemann Hypothesis (RH). The Extended Riemann Hypothesis (ERH) asserts the same for all Dirichlet L-series associated with characters.

Although my program is able to compute zeros of Dirichlet L-series without outside help, I used Michael Rubinstein's L-function calculator [3] to compute an initial approximation of the zeros I wanted, which were then refined (to 20 digits after the decimal point) using the PARI/GP calculator.

The character associated to the zeros in each (compressed) file given below is fully described in the header of that file. Only zeros with positive imaginary part are given.

Tables with approximate values of the first zeros of some Dirichlet L-series, last update made on August 7, 2007

Primitive characters:

Non-primitive characters:

Spectral analysis of the zeros of the zeta function

In 2010, inspired by the pair correlation of the differences between zeta zeros, in particular, by figure 13 of [4], I contemplated the possibility of predicting the location of the zeros of the zeta function (on the critical line) near an interval where all zeros had already been computed. To that end, and to make the predictions potentially more accurate, I worked not with the differences between consecutive zeros but with the differences between the zeros and known reference points.

Let γn be the (positive) imaginary part of the n-th zero of the zeta function on the critical strip; for example, γ1=14.134725.... For the range of values of n that we will be working with, the Riemann Hypothesis is known to be true, so the real part of the zeros will be 1/2. Let Gn be the n-th shifted Gram-point, which satisfies θ(t)=(n-3/2)π where θ(t) is the Riemann-Siegel theta function; for example, G1=14.517919.... Let unn-Gn and let vn=θ(γn)/π-(n-3/2); un is the difference between a zero and the corresponding reference point, and vn is the normalized difference. For example, u1=-0.383194... and v1=-0.050252.... Note that un/vn is approximately 2π/log(t/(2π)). The mean value of un, and that of vn, appears to be zero. At large height, that is, for large values of γn, when one is considering a relatively small interval the factor 2π/log(t/(2π)) will be almost constant, and so it will be almost the same to work with un or with vn. We use vn because it implies an average distance between normalized zeros of one.

Figure 13 of [4] suggests that if we compute a Discrete Fourier Transform (DFT) of a finite number of normalized differences of consecutive zeros of the zeta function on the critical line we get what appears to be spectral lines. Replacing the differences between consecutive zeros by the differences between the zeros and our reference points, and doing this for intervals centered at different locations we get the following figure.

We used 200 intervals per decade, and DFTs with 1281 points (the central point, plus 640 points on each side). Before performing the DFT we applied a Kaiser-Bessel window with α=3.0. This figure suggests that there exists a lot of underlying structure in the way the zeros of the zeta function are located. It even suggests a formula, very likely divergent, for the inverse of the κ(n) function introduced in [5].

Unfortunately, although it is possible to compute with reasonable precision the parameters of the more energetic "spectral lines" based on zero data on a given interval, given the plethora of "spectral lines" observed in the figure, our original goal of predicting the location of the zeros near a given interval turned out to give relatively poor results.

Pérez-Marco phenomenon

In an interesting preprint [6] Ricardo Pérez-Marco uncovered an unexpected anomaly in the distribution of the differences between zeros (not necessarily consecutive) of Dirichlet L-series on the critical line; in the special case of the zeta function, this anomaly was already known [7]. It appears that these differences have a somewhat smaller density when they are close to the imaginary parts of the zeros of the zeta function! Moreover, the differences between zeros of Dirichlet L-series also seem to "avoid" the imaginary parts of the zeta zeros! The following (very wide) figure illustrates what is going on for the zeta zeros. The interval between 0 and 200 was subdivided into 100000 small bins, and the number of differences between the first 10^6, 10^7, 10^8, and 10^9 zeros of the zeta function than fell in each bin was counted; thus, we are estimating the so-called pair correlation of the unscaled zeros. Each of the four curves presents the normalized counts (one corresponds to an uniform density) for the number of zeros given above. To reduce high-frequency fluctuations the data was smoothed using a triangular window and then down-sampled. The gray vertical lines pinpoint the location of the zeta zeros (see also figure 3 of [8]).

As can be seen in the figure, the dips in the density of differences coincide with the zeros of the zeta function. These dips become less pronounced when more zeta zeros are used to form the differences, as explained in [9].

References

 [1] H. Davenport, Multiplicative number theory, Graduate Texts in Mathematics, Vol. 74, Third Edition, 2000, Springer. [2] H. M. Edwards, Riemann's zeta function, 2001, Dover Publications, Inc. (first published in 1974 by Academic Press, Inc.). [3] M. Rubinstein, L-function calculator. [4] A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Mathematics of Computation, vol. 48, no. 177, pp. 273-308, January 1987. [5] J. Arias de Reyna and J. van de Lune, On the exact location of the non-trivial zeros of Riemann's zeta function, arXiv:1305.3844v2 [math.NT], June 4, 2013. [6] R. Pérez Marco, Statistics on Riemann zeros, arXiv:1112.0346v1 [math.NT], December 1, 2011. [7] M. V. Berry and J. P. Keating, The Riemann Zeros and Eigenvalue Asymptotics, SIAM Review, vol. 41, no. 2, pp. 236-266, 1999. [8] N. C. Snaith, Riemann Zeros and Random Matrix Theory, Milan Journal of Mathematics, vol. 78, no. 1, pp. 135-152, August 2010. [9] K. Ford and A. Zaharescu, Unnormalized differences between zeros of L-functions, arXiv:1305.2520v3 [math.NT], May 5, 2014.