Experimental evaluation of Euler sums

k=1
1. Introduction
2
1 1 m 1 + 2 + + k (?1)k+1(k + 1)?n m 1; n 1; k=1 ! 1 X 1 + + (?1)k+1 m (?1)k+1(k + 1)?n m 1; n 1; 1? 2 aa(m; n) = k k1 =1 X 1 + + 1 (k + 1)?n m 1; n 2; 1 + 2m h(m; n) = km k1 =1 ! X 1)k+1 (k + 1)?n 1 ? 21 + + (?km (m; n) = m 1; n 2; a m k1 =1 X 1 + 21 + + k1 (?1)k+1 (k + 1)?n m 1; n 1; h (m; n) = m m k1 =1 ! X 1)k+1 (?1)k+1(k + 1)?n m 1; n 1: 1 ? 21 + + (?km a (m; n) = m k=1 Explicit evaluations of some of the constants in these classes are presented with proofs in 6] and 7]. Table 1 contains a summary of these results (these include some facts already known to Euler). Here (t) = P1 k?t is the Riemann zeta function. Results for k=1 alternating sums are also given in 7]. Variants of the sums de ned above can be evaluated by using these results. Note for example that for all m 1; n 2 one can write ! 1 1 X 1 + + 1 k?n = X 1 + 1 + + 1 ?n 1 + 2m km 2m (k + 1)m (k + 1) k=0 k=1 1 1 X 1 + + 1 (k + 1)?n + X k?m?n 1 + 2m = km k=1 k=1 = h(m; n) + (m + n):
In response to a letter from Goldbach, Euler considered sums of the form 1 X 1 + 21 + + k1 (k + 1)?n m m k=1 for positive integers m and n. Euler was able to give explicit values for certain of these sums in terms of the Riemann zeta function. In a recent companion paper, Euler's results were extended to a signi cantly larger class of sums of this type, including sums with alternating signs. This research was facilitated by numerical computations using an algorithm that can determine, with high con dence, whether or not a particular numerical value can be expressed as a rational linear combination of several given constants. The present paper presents the numerical techniques used in these computations and lists many of the experimental results that have been obtained.
Abstract
D. H. Bailey: NAS Applied Research Branch, NASA Ames Research Center, Mo ett Field, CA 94035-1000, USA; dbailey@. J. M. Borwein: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada; jborwein@cecm.sfu.ca. Research supported by NSERC and the Shrum Endowment at Simon Fraser University. R. Girgensohn: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada; girgen@cecm.sfu.ca. Research supported by a DFG fellowship. 1
k=0
3 (4) + 1 2(2) = 11 4 ; sh(2; 2) = 2 2 360 2 (6) ? 1 (2) (4) + 1 3(2) ? 2(3) = 37 6 ? 2(3); sh(2; 4) = 3 3 3 22680 1 2(2) ? 1 (4) = 4 ; h(2; 2) = 2 2 120 8 (2) (4) + 2(3) = 2(3) ? 4 6 ; h(2; 4) = ?6 (6) + 3 2835 n?2 X sh(1; n) = h(1; n) = n (n2+ 1) ? 1 2 k=1 (n ? k) (k + 1); n?2 X sh(2; n) = n(n3+ 1) (n + 2) + (2) (n) ? n 2 k=0 (n ? k) (k + 2) n?2 k?1 X X (j + 1) (k + 1 ? j ) + h(2; n); +1 (n ? k) 3 k=2 j =1 2n2 + n + 1 (2n + 1) + (2) (2n ? 1) h(2; 2n ? 1) = ? n?1 2 X + 2k (2k + 1) (2n ? 2k); k=1 2n2 ? 7n ? 3 (2n + 1) + (2) (2n ? 1) sh(2; 2n ? 1) = 6 X 1 n?2(2k ? 1) (2n ? 1 ? 2k) (2k + 2) ?2 k=1 n?2 X 1 X (2k + 1) n?2?k (2j + 1) (2n ? 1 ? 2k ? 2j ); +3 j =1 k=1 for m + n odd: ! # " 1 m + n ? 1 (m + n) + (m) (n) h(m; n) = m 2 !# m+n " 2j ? 2! X 2j ? 2 (2j ? 1) (m + n ? 2j + 1) ? m?1 + n?1 j =1 if m is odd, ! # " 1 m + n + 1 (m + n) h(m; n) = ? m 2 !# m+n " 2j ? 2! X 2j ? 2 (2j 1) (m + n ? 2j + 1) + m?1 + n?1 j =1 if m is even. Table 1: Explicit Evaluations of Euler Sums 4
Experimental Evaluation of Euler Sums David H. Bailey, Jonathan M. Borwein and Roland Girgensohn June 24, 1994 Ref: Experimental Mathematics, vol. 3, no. 1 (1994), pg. 17{30
In response to a letter from Goldbach, Euler considered sums of the form 1 X 1 + 21 + + k1 (k + 1)?n : m m k=1 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 m = 1; n 2. Little has been done on this problem in the intervening years (see 5] 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 1 X 1 2 1 1 + 2 + + k k?2 = 4:59987 k=1 17 (4) = 17 4 4 360 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 a rmed it. (Unknown to us at that time, De Doelder had proved a related result in 1991 11] from which the above identity follows.) 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 con dence, whether a numerical value can be expressed as a rational linear combination of several given constants. These e orts produced even more empirical evaluations, suggesting broad patterns and general conjectures. Ultimately proofs were found for many of these experimental results. We will consider the following classes of Euler sums: 1 X 1 1 m 1 + 2 + + k (k + 1)?n sh(m; n) = m 1; n 2; k=1 !m 1 k+1 X 1 1 ? 2 + + (?1) sa(m; n) = (k + 1)?n m 1; n 2; k
ah(m; n) =
1 X
Similarly, let hk = 1 + 1=2 + + 1=k, and de ne h0 = 0. Then for all n 2 one can write 1 1 X X 2 1 1 2 1 + 2 + + k k?n = hk+1(k + 1)?n k=1 k=0 1 X 1 2 hk + k + 1 (k + 1)?n = k=0 1 1 X X 2 hk (k + 1)?n + 2 hk (k + 1)?n?1 = k=0 k=0 1 X + (k + 1)?n?2 = sh(2; n) + 2sh(1; n + 1) + (n + 2): 3