Computing zeros of Hecke ζ functions with Pari and ComputeL

This page contains some Pari scripts for computing zeros of the Dedekind ζ function of an algebraic number field and Hecke ζ functions associated to characters of finite order. In order to use these scripts you will need a copy of Pari and, for Hecke ζ functions, Tim Dokchister's ComputeL package.

A specific problem

The general-purpose scripts presented here were created in order to solve a specific problem: Given an algebraic number field K find a real number β such that ζ_K(s) has a simple zero close to ρ= 1/2 + βi and no other Hecke zeta function associated to a character of H(K) has a zero anywhere close. This problem is motivated by the study of elements with factorizations of distinct lengths in algebraic number fields. If we can find a solitary zero at point ρ, it implies that another complex function (that naturally arises in such investigations) must have a singularity there. This, in turn, implies irregularities in the distribution of algebraic integers with unique factorization length in K.

Computations showing that such a solitary zero exists for several algebraic number fields are quoted in the paper “On the distribution of algebraic numbers with prescribed factorization properties” (M. Radziejewski, submitted for publication). The precise results can be found here.

The scripts

The following scripts are available:

zeros-search.gp	Search for zeros of complex functions.
find-zetak-zeros.gp	Find zeros of the Dedekind ζ function (requires zeros-search.gp).
zetachi.gp	Compute the Dirichlet series coefficients of a Hecke ζ function associated to a character of the ideal class group.
verify.gp	Try to prove that a Hecke ζ function has a simple zero close to a given point or that it has no zero there. Uses some effective upper bounds for the second derivative of ζ(s, χ). Requires zetachi.gp and ComputeL.

The files contain some documentation and examples in the comments. Have fun!

There is also an automated set of scripts (a bunch of MS Windows XP® batch files and Pari scripts) that I used to prove the existence of solitary zeros. It is more difficult to use and more specific, hence probably less interesting, but I supply it to openly document the results quoted in the paper. If you do use it, please use caution: the batch files create and delete some files without warning!

All the files referred to on this page (as well as the page itself) can be downloaded from here.

© 2004 Maciej Radziejewski. The programs and scripts available from this page come WITHOUT ANY WARRANTY WHATSOEVER. You can use them at your own risk. If you copy any of the programs or scripts, please keep them together with this web page file.

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To send me an e-mail, you can click on my name (please modify the spam-protected address): Maciej Radziejewski.