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Applications of the PSLQ Algorithm

3. Applications of the PSLQ Algorithm There are a number of applications of integer relation detection algorithms in computational mathematics. One application is to analyze whether or not a given constant

, whose value can be computed to high precision, is algebraic of some degree n or less. This can be done by first computing the vector

) to high precision and then applying an integer relation algorithm to the vector x. If a relation is found, this integer vector is precisely the set of coefficients of a polynomial satisfied by

. Even if a relation is not found, the resulting bound means that

cannot possibly be the root of a polynomial of degree n, with coefficients of size less than the established bound. Even negative results of this sort are often of interest.

We have performed several computations of this type [6]. These computations have established, for example, that if Euler's constant

satisfies an integer polynomial of degree 50 or less, then the Euclidean norm of the coefficients must exceed

. Computations of this sort have also been applied to study a certain conjecture regarding the Riemann zeta function. It is well known [10] that

These results have led some to suggest that

might also be a simple rational or algebraic number. Unfortunately, integer relation calculations [3] have established that if satisfies a polynomial of degree 25 or less, then the Euclidean norm of the coefficients must exceed .

4. Euler Sums

In response to a letter from Goldbach, Euler considered sums of the form

Euler was able to give explicit values for certain of these sums in terms of the Riemann zeta function. For example, Euler found an explicit formula for the case

. Little progress has been made on this problem in the intervening years, although special cases of Euler's results have been rediscovered numerous times (see [7] for some references). In April 1993, Enrico Au-Yeung, an undergraduate at the University of Waterloo, brought to the attention of one of us the curious fact that

based on a computation to 500,000 terms. This author's reaction was to compute the value of this constant to a higher level of precision in order to dispel this conjecture. Surprisingly, a computation to 30 and later to 100 decimal digits still affirmed it.

Intrigued by this empirical result, we computed numerical values for several of these and similar sums, which we have termed Euler sums. We then analyzed these values by a technique we will present below that permits one to determine, with a high level of confidence, whether a numerical value can be expressed as a rational linear combination of several given constants. These efforts produced even more empirical evaluations, suggesting broad patterns and general conjectures. Ultimately proofs were found for many of these experimental results. We will consider here the following two classes of Euler sums. Some other classes are considered in [4].

Explicit evaluations of some of the constants in these classes are presented with proofs in [8] and [9].

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