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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.

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.

Reviewer: Hermanus J. J. te Riele (Amsterdam)

### MSC:

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 |

### Keywords:

lattice approximation; computational number theory; algorithm; integral relations; constants; Euler’s constant; imaginary part of the first complex zero; Riemann’s zeta-function evaluated at 3; multiprecision computations### Citations:

Zbl 0616.10027
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\textit{D. H. Bailey} and \textit{H. R. P. Ferguson}, Math. Comput. 53, No. 188, 649--656 (1989; Zbl 0687.10002)

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### References:

[1] | David H. Bailey, The computation of \? to 29,360,000 decimal digits using Borweins’ quartically convergent algorithm, Math. Comp. 50 (1988), no. 181, 283 – 296. · Zbl 0641.10002 |

[2] | David H. Bailey, Numerical results on the transcendence of constants involving \?,\?, and Euler’s constant, Math. Comp. 50 (1988), no. 181, 275 – 281. · Zbl 0646.10002 |

[3] | D. H. Bailey, ”A high performance FFT algorithm for vector supercomputers,” Internat. J. Supercomput Appl., v. 2, 1988, pp. 82-87. |

[4] | J. M. Borwein and P. B. Borwein, The arithmetic-geometric mean and fast computation of elementary functions, SIAM Rev. 26 (1984), no. 3, 351 – 366. · Zbl 0557.65009 |

[5] | J. M. Borwein & P. B. Borwein, Pi and the AGM–A Study in Analytic Number Theory and Computation Complexity, Wiley, New York, 1987. · Zbl 0611.10001 |

[6] | J. M. Borwein & P. B. Borwein, ”Ramanujan and Pi,” Sci. Amer., v. 258, 1988, pp. 112-117. · Zbl 0652.10019 |

[7] | J. V. Drazil, Quantities and Units of Measurement: A Dictionary and Handbook, Mansell Publishing, Ltd., London, 1983. |

[8] | Mitchell J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), no. 1, 25 – 52. · Zbl 0509.58037 |

[9] | H. R. P. Ferguson and R. W. Forcade, Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 912 – 914. · Zbl 0424.10021 |

[10] | Helaman Ferguson, A short proof of the existence of vector Euclidean algorithms, Proc. Amer. Math. Soc. 97 (1986), no. 1, 8 – 10. · Zbl 0592.10027 |

[11] | Helaman R. P. Ferguson, A noninductive \?\?(\?,\?) algorithm that constructs integral linear relations for \?\?-linearly dependent real numbers, J. Algorithms 8 (1987), no. 1, 131 – 145. · Zbl 0649.10021 |

[12] | H. R. P. Ferguson, A New Integral Relation Finding Algorithm Involving Partial Sums of Squares, Brigham Young University preprint, Sept. 1987. |

[13] | H. R. P. Ferguson, ”A new integral relation finding algorithm involving partial sums of squares and no square roots,” Abstracts Amer. Math. Soc., v. 9, 1988, p. 214. |

[14] | I. J. Good, On the masses of the proton, neutron and hyperons, J. Roy. Naval Sci. Service 12 (1957), 144. |

[15] | Ravi Kannan and Lyle A. McGeoch, Basis reduction and evidence for transcendence of certain numbers, Foundations of software technology and theoretical computer science (New Delhi, 1986) Lecture Notes in Comput. Sci., vol. 241, Springer, Berlin, 1986, pp. 263 – 269. · Zbl 0616.10027 |

[16] | J. Håstad, B. Helfrich, J. Lagarias, and C.-P. Schnorr, Polynomial time algorithms for finding integer relations among real numbers, STACS 86 (Orsay, 1986) Lecture Notes in Comput. Sci., vol. 210, Springer, Berlin, 1986, pp. 105 – 118. · Zbl 0606.68033 |

[17] | A. M. Odlyzko and H. J. J. te Riele, Disproof of the Mertens conjecture, J. Reine Angew. Math. 357 (1985), 138 – 160. · Zbl 0544.10047 |

[18] | B. Robertson, ”Wyler’s expression for the fine-structure constant \( \alpha \),” Phys. Rev. Lett., v. 27, 1971, pp. 1545-1547. |

[19] | Paul N. Swarztrauber, Multiprocessor FFTs, Proceedings of the international conference on vector and parallel computing — issues in applied research and development (Loen, 1986), 1987, pp. 197 – 210. · Zbl 0624.65146 |

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