Conventions, Habits and Notations

We have used FORM and TFORM for all symbolic calculations. It should be rather clear why, but just in case you have not noticed: there is an overlap in authors. Also, FORM can handle really big expressions fast. Some of the calculations involved processing several TeraTerms (1 TeraTerm = 1000000000000 terms). Another advantage is that (T)FORM is freely available.

All major edit work was done with the STedi editor which is a folding editor. This allows good structure inside files. Hence the libraries consist usually of one file with several procedures. The editor and its sources are also freely available. You may find fold information in some files. This is usually of the type

    *--#[ name : commentary
        lines of code or text or commentary
    *--#] name :

When the fold is closed all the above (no matter how many lines are inside) will look like

    *--## name : commentary

The star in column 1 guarantees that the fold lines are seen as commentary by FORM (in for instance TeX one would use the % sign). More information can be found in the manual of the editor of which there exist an on-line version and printable versions.

During the project we changed notations with the constants. The modern version is:

   habcd = H(-a,-b,-c,-d)
   zazbzczd = H(a,b,c,d) = Z(a,b,c,d)
   a3a5a6a7 = A(3,5,6,7)

in which a,b,c,d belong to the set 0-9,A-Z when the h is involved and for the z sums a,b,c,d are positive integers. In the case of the h sums the alphabetic characters are A-Z = 10-35. Hence hF3 = H(-15,-3).

It is not 100% excluded that there are still some files with older conventions. The files of the original summer program use an even older convention in which a number of constants had been, more or less, randomly selected as basis elements and just been given a name.

We distinguish between S-sums, Z-sums and H-functions. The H-functions are harmonic polylogarithms in one. When all indices are positive they are identical to the Z-sums in infinity. All calculations have been done for H-functions in one. This turned out to be most convenient. There are conversion routines in the programs section. At times we convert to either Z-sums or S-sums inside a procedure, because we use some formulas that are simpler in that notation.

More conventions, S-sums, Z-sums and H-functions are explained in the paper.

We use a naming convention of the files that makes it possible to put all results together in a single directory without having name conflicts. This is particularly handy when one likes to have a single library directory with either .tbl or .prc files. If this directory is part of the FORMPATH variable in the shell you use, FORM will find these files automatically.

The name conventions are:

• mzv stands for Multiple Zeta Value, or non-alternating sums.
• alt stands for Euler sums, or alternating sums.
• The number following alt or mzv is the weight.
• If nothing follows this number, this means that the file contains complete results (all depths) for this weight over the rational numbers.
• The number can be followed by the letter r for rational arithmetic including products of lower weight objects, the letter t for rational arithmetic but without the product over lower weight objects, or the letter m for arithmetic modulus the number 2147479273 dropping the products of lower weight objects.
• If nothing follows the letter m, the file contains complete results (all depths) for this weight in modular calculus (dropping the products of lower weight objects).
• If the letters m, t or r are followed by a number, this number indicates the maximum value of the depth.

In the case of the MZVs we have omitted all objects that can be transformed into lower depth objects by means of the duality relation. For even weights and depth = weight/2 of an object and its dual we keep the one that comes first lexicographically (largest number of leading zeroes in integral notation).

Of each of the above types of data sets there are 4 varieties:

• .log Here usually the file has a different name, but this is the log file of the run as it actually took place. This file is in text format.
• .prc This is the result of editing the log file and making it into the definition of a table. This file is in text format.
• .sav A FORM binary file that allows for reading the combined results of the run without running into compilation restrictions (could happen otherwise with files of several Gbytes). These files are in principle portable between 64-bits systems and 32-bits systems.
• .tbl A FORM binary file which allows a database-like access to the table elements. This is the preferred mode and gives very fast access. Unfortunately only usable on 64-bits systems.

Examples:

mzv20.tbl is the tablebase file with all MZVs of weight 20. All, means up to the divergent ones and the ones that can be obtained by duality.

mzv27m7.prc is the procedure file that defines the table mzv27m7 and all its elements, which are MZVs of weight 27 and a depth of maximally 7.

alt20r4.sav is the FORM binary file with the complete results of the run at weight 20 with a maximal depth of 4 in rational arithmetic.