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Application of PSLQ to Euler Sums

6. Application of PSLQ to Euler Sums The present application of Euler sum constants is well suited to analysis with integer relation algorithms. We will present but one example of these computations. Consider

Based on experience with other constants, we conjectured that this constant satisfies a relation involving homogeneous combinations of

and

, where

denote s the polylogarithm function. The numerical values of these constants, to 50 decimal digits, are as follows:

The set of terms involving these constants with degree five (see section 7) are as follows:

. When

is augmented with this set of terms, all computed to 135 decimal digits accuracy, and the resulting 9-long vector is input to the PSLQ algorithm, it detects the relation

, at iteration 390. Solving this relation for

, we obtain the formula

(recall that

). When the relation is detected, the minimum and maximum y vector entries are

and

, respectively. Thus the confidence level of this detection is on the order of

, indicating a very reliable detection.

Although 135-digit input values and 150-digit working precision were used by us when this relation was originally detected, the fact that the maximum y-vector entry is only

at detection implies that such high levels of numeric precision are not required in this case. Indeed, the above relation can be successfully detected using only the 50-digit input values listed above and 50-digit working precision when performing the PSLQ algorithm.

Many special cases of the proven results listed in Table 1 were first obtained using the experimental method presented in sections 2 through 4. In addition, we have obtained a number of experimental results for which formal proofs have not yet been found. Tables 2 and 3 list some of these experimental identities. Others can be found in [4].

It should be emphasized that the results in Tables 2 and 3 are not established in any rigorous mathematical sense by these calculations. However, in each case the ``confidence level'' (see section 3) of these detections is less than

, and in most cases is in the neighborhood of

Table 1: Experimentally Detected Results

In many other cases we were not able to obtain a formula for the Euler sum constant explicitly in terms of values of the Riemann zeta, logarithm and polylogarithm functions, but we were able to obtain relations involving two or more Euler sum constants of the same degree (where by ``degree'' we mean m + n, where m and n are the indices of the constant). Some of these relations are shown in Table 3. This is not a complete list; we have obtained numerous other relations of this type. The ``confidence level'' of each of these relations is smaller than

. The uniqueness of each of these relations was checked by repeating the run with one fewer constant input to PSLQ (there should be no relation detected when this is done).

Table 2: Experimentally Detected Relations

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