Formulae for some classical constants Alexandru Lupaş (to appear in Proceedings of ROGER-000 The goal of this paper is to present formulas for Apéry Constant, Archimede s Constant, Logarithm Constant, Catalan s Constant Such formulas are useful for high precision calculations frequently appearing in number theory Also, one motivation for computing digits of some constants is that these are excellent test of the integrity of computer hardware and software The Apery s Constant is defined as The Apéry s Constant ζ(3 ζ(3 = = 3 = = 0056903595948539973865449990764986934049 The designation of ζ(3 as Apéry s constant is new but well-deserved In 979, Roger Apéry stunned the mathematical world with a miraculous proof that ζ(3 is irrational (see [4], [6] The above expression is very slow to converge Another formula is Roger Apéry ([4]-979 ( ζ(3 = 5 ( ( = 3 which was the starting point of Apéry s incredible proof of the irrationality of ζ(3 Since 980, many formulas for ζ(3 were discovered Ro-Ger University, Calea Dumbrăvii 8-3, 400-Sibiu, E-Mail : lupas@jupitersibiuro
For instance TAmderberhan ([]-996 proved that ( (3 with ζ(3 = ( (56 3 + 5 ( (! 3 4 = ( (3! ζ(3 = 8 ( (! (! A( (4! = 3 B( A( = 565( 4 + 3878( 3 + 376( + +60( + 040 B( = (3 (3 TAmdeberhan and Doron Zeilberger ([]-996, ( [3]-997 (4 where (A Lupaş (5 with ζ(3 = 375 35 + 650 ζ(3 = ( Q( ( = 5 5 Q( = 05 60 + 3 ( P( ( = 5 5( + 5 P( = 4760 4 + 800 3 + 9505 + 590 + 67 The proof of (5 was performed by means of WZ-method (see [7],[9] also [],[3] followed by a Kummer transformation Let us remind that a discrete function a(n, defined for n, {0,, } is a(n +, called a Closed Form =(CF in two variables when the ratios, a(n, a(n, + are both rational functions A pair (F, G of CF functions which a(n, satisfy (6 F (n +, F (n, = G(n, + G(n, is a so-called WZ- form There are many investigations connected with such WZ-forms (today nown as WZ-theory : see [], [3],[7],[9] In the 990 s, the WZ theory has been the method of choice in resolving conjectures for hypergeometric identitiesthe ey of the WZ-theory is the following theorem
Theorem (D Zeilberger-[7], 993 For any WZ-form (F,G ( G(, 0 = F ( +, + G(,, whenever either side converges Let us remar that all formulas which are listed in (-(5are of the form (7 ζ(3 = K + K = ( a, a > 0 where K, K are some real constants Supposing that we have also another representation, namely (8 ζ(3 = K + K ( b, b > 0 = we shall say that (7 (8 iff ζ n a n = O(ζ n, b n = O(η n with lim = 0 n η n Formula a n = O(ζ n ( a n = O ( 4 n n n ( a n = O ( 7 n n (3 a n = O ( 64 n n (4 a n = O ( 04 n n (5 a n = O ( 04 n n3 n Therefore (5 (4 (3 ( ( 3
GFee and Simon Plouffe used (4 in their evaluation of ζ(3 with 50,000 digits (see [3] Later, by means of the same formula (4 BHaible and TPapaniolau computed ζ(3 to,000,000 digits Open problem : If T n (x = cos (n arccos x is the Chebychev s polynomial of the first ind, and (see (5 P (j = 4760j 4 + 800j 3 + 9505j + 590j + 67 S = ( j P (j ( 650 j= j 5 j 5(j j + 5 A n = ( n n n + 4 S T n (3 = n + B n = 375 + A 35 n then prove or disprove that ζ(3 B n = O ( 5939 n n 3 n, (n The following proposition generalize an identity proved by RAp ery ( x = 0, [4] Theorem For x 0 n = ( + x 3 = 5 + n = n = ( 3 ( +x ( +x ( 3 ( n++x A more general function than ζ(s is ζ(s, x = ( ( n+x x(3 + x + 5( + x ( + x s, Re(s >, x > 0 Corollary If x > 0, then ζ(3, x + = 5 ( ( ( 3 +x +x ( x(3 + x 5( + x 4
The constants ln, G, π Theorem 3 Let x > 0, y > 0 Then (9 ( ( (x (x + y = ( 3 = y ( (y + 0( + y + (6x + ( + y + x 6 ( (y + (x + y + + y Proof Define A(j = A(j, x, y = y ( j 6 j ( (y + j 3 ( (y + j and G(n, = λ(n, = ( n( (x n (x + y n+ λ(n, ja(j, = j y+j (y + j + x + nλ(n, j + A(j, = j + Further F (n, = λ(n,j A(j, = j (y+j (n + x + 3y + 3j + λ(n, j + A(j, = j + If G(n, and F (n, are defined as above, then F (n +, F (n, = G(n, + G(n,, that is (F, G is a WZ-form By applying WZ-method we find (9 Further, by G is denoted the Catalan constant G which is defined as G = β( = ( ( + = 09596554, 5
( β(x := being the Dirichlet s beta function ( + x Some particular cases of (9 are x y Identity a The order of a n (n π = 4 ( ( a = 40 +6+ ( 4 (4+ O ( 64 n n ln = 3 ( a = 6 ( 5+ (+ O ( 4 n n 3 G = ( a = (Catalan s Constant 8 (40 4+3 3 ( 4 ( 64( O ( 4 n n π = ( a 3 ( 4 9+ O ( 3( = 4 n n 3 The mathematicians have assumed that there is no shortcut to determining just the n th digit of π Thus it came as a great surprise when such a scheme was in 997 discovered [6] (see ( from below This formula was discovered empirically, using months of PSLQ computations, see [8] This is liely the first instance in history that a signifiant new formula for π was discovered by a computer In the following, we present an identity which gives a proof of ( Proposition If z + <, r C, then z z π + 4 arctan z + ( + 8r ln = (4 + 8r + z ( + z 8+ 8 + 6 (0 8r ( + 8r ( + r ( + z 8+ 8 + ( + z 8+4 8 + 4 ( + z 8+6 8 + 6 6 4r ( + z 8+3 8 + 3 6 ( + r 6 + r 6 ( + z 8+5 8 + 5 ( + z 8+7 8 + 7 6 6 6
A proof of the above asertion is given in [5] If in (0 we select (z, r = (0, then π = Liewise, using the identity one finds π = ( 4 8 + + 8 + 3 + 8 + 4 (8 + 7 6 arctan + arctan z = arctan t, t := + z z, (z < ( 8 + + 8 + + 8 + 3 (8 + 5 (8 + 6 4(8 + 7 6 Let us remar that for (z, r = (0, 0 we find from (0 the remarable BBP formula (attributed to David Bailey, Peter Borwein and Simon Plouffe for π, that is ( π = ( 4 8 + 8 + 4 8 + 5 8 + 6 6 Also when (z, r = (0, r, r C one finds another formula for π, namely = ( 4 + 8r 8 + 8r 8 + π = 4r 8 + 3 + 8r 8 + 4 + r 8 + 5 + r 8 + 6 + r 8 + 7 6 which was discovered by Victor Adamchi and Stan Wagon in their interesting HTML paper [] References [] V Adamchi, S Wagon, Pi : A 000 -Year Search Changes Direction, see http : //wwwmemberswricom/victor/articles/pi/pihtml [] T Amderberhan, Faster and faster convergent series for ζ(3, Electronic J Combinatorics 3 (996 [3] T Amderberhan, D Zeilberger, Hypergeometric series acceleration via the WZ method, Electronic J Combinatorics (Wilf Festschrifft Volume 4 ( (997 7
[4] R Apéry, Irrationalité de ζ( et ζ(3, Astérisque 6 (979-3 [5] DH Bailey,The Computation of Pi to 9,360,000 Decimal Digits Using Borweins Quartically Convergent Algorithm, Mathematics of Computation, Jan (988 83-96 [6] DH Bailey, PB Borwein, S Plouffe, On the Rapid Computation of Various Polylogarithmic Constants, Mathematics of Computation 66 (997 903-93 [7] DHBailey, JM Borwein, SPlouffe, The Quest for Pi, The Mathematical Intelligencer 9( (997 50-57 [8] DH Bailey, DJ Broadhurst, Parallel Integer Relation Detection : Techniques and Applications, Mathematics of Computation (to appear, see also http://frontmathucdavisedu/mathna/9905048 [9] JM Borwein, DM Bradley, RECrandall, Computational strategies for the Riemann zeta function, CECM preprint 98:8 (998 [0] P Borwein, An Efficient Algorithm for the Riemann Zeta Function, (to appear [] DJ Broadhurst, Polylograithmic ladders, hypergeometric series and the ten milionth digits of ζ(3 and ζ(5, (Preprint (998 [] RE Crandall,Topics in Advanced Scientific Computation, Springer, New Yor, 995 [3] S Finch, Favorite Mathematical Constants, http://wwwmathsoftcom/asolve/constant/apery/aperyhtml [4] S Finch, The Miraculous Bailey-Borwein-Plouffe Pi Algorithm, (forthcoming, http : //wwwmathsof tcom/asolve/plouf f e/plouf f ehtml [5] A Lupaş, Some BBP-Functions, DVI- paper, (see [3]-[4] [6] A van der Poorten, A Proof that Euler missed Apéry s proof of the irrationality of ζ(3, MathIntelligencer (979, 96-03 [7] D Zeilberger, Closed Form (pun intended!, Contemporary Mathematics 43 (993 579-607 [8] HS Wilf, D Zeilberger, Rational functions certify combinatorial identities, Jour AmerMathSoc 3 (990 47-58 [9] H Wilf, Accelerated series for universal constants, by the WZ method, DiscrMathTheoretCompSci 3 (999 89-9 8