This set of scripts serves the purpose of studying the multiplicities of zeros
of some Hecke zeta functions. It is only for Ms Windows systems.

For a given algebraic number field K and a real number ImRho we try to check
if zeta_K(s) has a simple zero close to rho = 1/2 + ImRho*I and to show that
no other Hecke zeta function associated to a character of H(K) has a zero
anywhere close.

To use it, first set the parameters in the file "parameters.txt", like this:

fpol = x^2+65; \\ The defining polynomial of the field.
ImRho = 1.05325893699922446326153868; \\ Imaginary part of suspected zero of zeta_K.

Then you may need to edit the batch files present in this directory to correct
the path to Pari. Moreover, the script file "verify-zetachi-iterate.gp"
creates a batch file on the fly - you will need to modify the path to Pari in
"verify-zetachi-iterate.gp", too.

After completing these steps you are ready to run the batch file "start.bat".
The results will be written to a file "report.txt":

Defining polynomial - input parameter
Im rho              - input parameter
Funtional equation error - A diagnostic value from the ComputeL package, should be close to 0.
Simple zero - If this value is less than 1, we can claim a simple zero is located within
              a certain distance r from rho.
              Please bear in mind that computational errors may affect results,
              so it is best if this value is close to 0.
Distance    - The value of r, see above.
Competition - The lower bound for the distance of the closest zero of zeta(s, chi), chi!=chi0.
Contradiction - A diagnostic value, the ratio of lower to upper bound for zero distance.
                Should be less than 1.

The same results will be written without descriptions to a table in "report_tabular.txt".
Please refer to the file verify-common.gp for the description of the method.

It is rather awkward to have so many scripts calling each other.
However, I was not sure if I can use ComputeL repeatedly within one Pari session
for multiple zeta functions. It is safer to restart Pari every time.
We need to iterate through the characters of the class group, store temporary
variables, etc. Hence so many files and scripts creating batch files calling
other scripts...

You can download the scripts from my homepage http://amu.edu.pl/~maciejr .
Please let me know if there are any problems or errors in the scripts and, of
course, if you find these scripts useful! My e-mail is maciejr@amu.edu.pl .

Maciej Radziejewski.