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.
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
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 following scripts are available:
||Search for zeros of complex functions.|
||Find zeros of the Dedekind ζ function
|zetachi.gp||Compute the Dirichlet series coefficients of a Hecke ζ function associated to a character of the ideal class group.|
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
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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