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Numerical results on relations between fundamental constants using a new algorithm. (English) Zbl 0687.10002
A new algorithm (found by the second author) for finding integral relations among a given set of real numbers is described and applied to several sets of real numbers. In particular, it is established that none of the following constants satisfies a simple, low degree polynomial equation: \(\gamma\) (Euler’s constant), \(\log \gamma\), \(\log \pi\), \(\rho_1\) (the imaginary part of the first complex zero of the Riemann zeta-function), \(\log \rho_1\), \(\zeta(3)\) (Riemann’s zeta-function evaluated at 3), and \(log \zeta(3)\). Moreover, minimum Euclidean norms are listed for any integer relation which could possibly be satisfied by vectors consisting of powers of the above numbers. The algorithm is also applied to Feigenbaum’s constant, and to two constants of fundamental physics. However, these constants are only known to very moderate precision (derived from experiments), so that the results found for these constants are much weaker than those obtained for \(\gamma\), \(\log \gamma\), etc.
Multiprecision computations are used to run the algorithm (with an accuracy of up to 768 decimal digits). The multiplication procedure is based on a fast complex FFT routine. Much care is devoted to the reliability of the computational results. The computations were performed on two different computer systems (a Cray-2 supercomputer and a Silicon Graphics IRIS 4D workstation) with different programs on the two computers. The new algorithm is much faster than a previous algorithm of Forcade and Ferguson (by a factor of at least 300 for a given example).
Similar recent work by R. Kannan and L. A. McGeoch [Lect. Notes Comput. Sci. 241, 263–269 (1986; Zbl 0616.10027)] is briefly mentioned: this utilizes the Lovász basis reduction algorithm to obtain bounds on any polynomial that could be satisfied by \(e-\pi\) and \(e+\pi\). It would be interesting to see how Kannan and McGeoch’s attack compares with the new algorithm studied in the present paper.

11-04 Software, source code, etc. for problems pertaining to number theory
11Y16 Number-theoretic algorithms; complexity
11J81 Transcendence (general theory)
11Y35 Analytic computations
11Y60 Evaluation of number-theoretic constants
Zbl 0616.10027
Full Text: DOI
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