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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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