D. Bailey, Numerical results on the transcendence of constants involving , e, and Euler's constant, Math. Comp. 50 (1988), 275--281.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....zeta function, the divisor function d(n) etc. 15] seems to be more mysterious and more difficult to compute than . For example, quadratically convergent iterations are known [6, 8, 21] for , but none are known for fl. Also, is transcendental, but it is not even known if fl is irrational [3]. If fl = p=q is rational, then q 10 15000 . This result follows from a computation [10] of the regular continued fraction expansion for fl. There may be an analogy with i(3) Ap ery [2, 17] proved i(3) irrational, using the series i(3) 5 2 1 X k=1 ( Gamma1) k Gamma1 k k (2k) k 3 ....

D. Bailey, Numerical results on the transcendence of constants involving , e, and Euler's constant, Math. Comp. 50 (1988), 275--281.

....zeta function, the divisor function d(n) etc. 15] seems to be more mysterious and more di cult to compute than . For example, quadratically convergent iterations are known [6, 8, 21] for , but none are known for . Also, is transcendental, but it is not even known if is irrational [3]. If = p=q is rational, then q 10 15000 . This result follows from a computation [10] of the regular continued fraction expansion for . There may be an analogy with (3) Ap ery [2, 17] proved (3) irrational, using the series (3) 5 2 1 X k=1 ( 1) k 1 k k (2k) k 3 ; and, in ....

D. Bailey, Numerical results on the transcendence of constants involving , e, and Euler's constant, Math. Comp. 50 (1988), 275-281.

....numerical bound) One can show, on using a working precision that is only slightly higher than that of the input data, that these bound results obtained from computer runs are reliable. Results of this type, supporting conjectures about transcendence of certain constants, are to be found in [2, 4]. See also Section 2.3 on minimal polynomials. For example, Bailey and Ferguson [4] used PSLQ to establish, that if Euler s constant fl satisfies an integer polynomial of degree 50 or less, then the Euclidean norm of the coefficients must exceed 7 Delta 10 17 . In the context of these ....

D. Bailey, Numerical results on the transcendence of constants involving pi, e, and Euler's constant. Math. Comp. 50 (1988), 275--281.

....relation is found, these algorithms also produce bounds within which no relation can exist, which results are of interest by themselves. Clearly such analysis can be applied to any constant that can be computed to suciently high precision. The author has performed some computations of this type [4, 5], and others are in progress. Some recent results include the following: if satis es an integer polynomial of degree 50 or less, then the Euclidean norm of the coecients must exceed 7 10 17 ; if Feigenbaum s constant satis es an integer polynomial of degree 20 or less, then the Euclidean ....

Bailey, D. H., \Numerical Results on the Transcendence of Constants Involving , e, and Euler's Constant", Mathematics of Computation, vol. 50 (Jan. 1988), p. 275 { 281.

....15] These algorithms were shown to be polynomial time in the logarithm of the size of a smallest relation. They were not shown to be polynomial in the dimension. Since then, other related algorithms for finding relations for real vectors have appeared in [8] 16] 17] 18] For example, [5] reports on a computer implementation of [16] The sequence including [23] HJLS) 2] and [1] PSLQ) 3] a concise statement of PSLQ) and [35] a stable variation of HJLS) will be discussed below. These algorithms all depend upon an orthogonal decomposition of some kind. See [21] for a list ....

D. H. Bailey, Numerical results on the transcendence of constants involving ß, e, and Euler's constant, Mathematics of Computation 50 (January 1988), no. 181, 275 -- 281.

....of integers a i , these integers are the coefficients of a polynomial satisfied by ff. Even if no such relation is found, these algorithms also produce bounds within which no relation can exist, which results are of interest by themselves. The author has performed some computations of this type [3, 5], and others are in progress. Some recent results include the following: if fl satisfies an integer polynomial of degree 50 or less, then the Euclidean norm of the coefficients must exceed 7 Theta 10 17 ; if Feigenbaum s ffi constant satisfies an integer polynomial of degree 20 or less, then ....

D. H. Bailey, "Numerical Results on the Transcendence of Constants Involving ß, e, and Euler's Constant", Mathematics of Computation, vol. 50 (Jan. 1988), p. 275 -- 281.

....complexity [16] An unfortunate feature of the above algorithms that severely limits their practical application is that they require enormously high numeric precision (and correspondingly long run times) in order to obtain meaningful results. For example, one of the calculations cited in [2] established that Euler s constant fl cannot satisfy any algebraic polynomial of degree eight or less and with coefficients of size 10 9 or smaller. This calculation, which employed one of the above algorithms, required 6,144 digit precision and two hours CPU time on a Cray 2 supercomputer. Such ....

....versions, it still requires multiprecision arithmetic in step 3 above to obtain strong results for n greater than three or four. For this purpose a package of high performance multiprecision arithmetic routines was employed. These routines are similar to those described in detail in [1] and [2]. The main difference in the routines used for this application is the incorporation of an even faster complex FFT routine [3] at the heart of the multiprecision multiplication procedure. This new FFT, which employs a radix 4 version of an algorithm suggested by Swarztrauber [19] is presently the ....

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Bailey, D. H., "Numerical Results on the Transcendence of Constants Involving ß, e, and Euler's Constant", Mathematics of Computation, Vol. 50, No. 181 (Jan. 1988), p. 275 - 281.

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