Ramanujan and Euler's Constant

Transcription

1 Proceedings of Symposia in Applied Mahemaics Volume 00, 0000 Ramanujan and Euler's Consan RICHARD P. BRENT Absrac. We consider Ramanujan's conribuion o formulas for Euler's consan. For eample, in his second noebook Ramanujan saes ha (in modern noaion) ( ) k = ln + + o() nk as!. This is known o be correc for he case n =, bu incorrec for n > 2. We consider he case n = 2. We also sugges a dieren, correc generalizaion of he case n =.. Inroducion Ramanujan gave many beauiful formulas for and =. Euler's consan = 0 () = 0: : : :, which occurs in many well-known formulas involving he Gamma funcion, he Riemann zea funcion, he divisor funcion d(n), ec. [5], seems o be more myserious and more dicul o compue han. For eample, quadraically convergen ieraions are known [6, 8, 2] for, bu none are known for. Also, is ranscendenal, bu i is no even known if is irraional [3]. If = p=q is raional, hen q > This resul follows from a compuaion [0] of he regular coninued fracion epansion for. There may be an analogy wih (3). Apery [2, 7] proved (3) irraional, using he series (3) = 5 2 ( ) k (2k)!k 3 ; and, in Chaper 9 of his Noebooks, Ramanujan gives several similar series, some 99 Mahemaics Subjec Classicaion. Primary 0A60, 33C0, 4A60; Secondary 33-0, 33-03, 33E99, 40A25, 65D20. To appear in Mahemaics of Compuaion 943{993. rpb39 ypese using AMS-LATEX c993 American Mahemaical Sociey /00 $.00 + $.25 per page

2 2 R. P. BRENT involving (3). Ramanujan [4, I, p. 252] rediscovered Euler's formula (3) = (k + ) 2 ; P k where = j= =j is a Harmonic number. Harmonic numbers also occur in formulas involving. For eample, he well-known resul () H n = ln n + + O(=n) as n! is ofen used o give an alernaive deniion of. 2. Ramanujan's Papers and Noebooks Ramanujan published one paper [8] specically on. In i he generalizes an ineresing series of Glaisher: = (2k + ) (k + )(2k + ) : Because he generalizaions all involve he Riemann zea funcion or relaed funcions, hey are no convenien for compuaional purposes. Much of Ramanujan's work was no published during his lifeime, bu was summarized in his Noebooks. These were rs prined in facsimile [20], and edied ediions have since been published by Bernd [4]. Scanning he Noebooks, we nd many occurrences of. Owing o space limiaions, we concenrae on Chaper 4, Enry 9, Corollaries {2 [4, I, p. 98], because hese are poenially useful for compuing. Corollary is (in modern noaion) (2) ( ) k k k = ln + + o() as!. In fac, Euler showed he more precise resul [4, II, p. 67], Z ( ) k k e e (3) ln = k d = O ; and his has been used by Sweeney [22] and ohers [5, 9] o compue Euler's consan (one has o be careful because of cancellaion in he series). In Ch. 2, Enry 44(ii) [4, II, p. 68], Ramanujan correcly saes ha he error erm O(e =) in (3) is beween e =( + ) and e =. 2.. A Generalizaion. Ramanujan's Corollary 2 [4, I, p. 98] is ha ( ) k (4) = ln + + o() nk for n > 0. We assume ha n is a ed posiive ineger. Clearly (4) generalizes (2), which is jus he case n =.

3 RAMANUJAN AND EULER'S CONSTANT 3 Bernd [4, I, p. 98], using a resul of Olver [6, E. 8.4, p. 309], shows ha (4) is false for n > 2, because he funcion dened by he lef side of (4) changes sign inniely ofen and grows eponenially large as!. Bernd leaves he case n = 2 open. In fac, (4) is rue if n = 2. Theorem below gives an eac epression for he error in (4) as an inegral involving he Bessel funcion J 0 (), and Corollary deduces an asympoic epansion. The eac epression for n = 2 is a special case of a formula given by Luke [3, p. 48] and [, formula..20]. However, he connecion wih Ramanujan does no seem o have been noiced before Avoiding Cancellaion. In Chaper 3, Enry 2, Corollary 2, Ramanujan saes ha he sum on he lef side of (2) can be wrien as e k : This is easy o prove [4, I, pp. 46{47]. Thus, (2) gives (5) k X k = ln + + o(): This is more convenien han (2) for compuaion, because here is no cancellaion in he series when > 0. In Secion 4 we indicae how Ramanujan migh have generalized (5) in much he same way ha he aemped o generalize (2). 3. Ramanujan's Corollary for n = 2 The following resul [] shows ha (4) is valid for n = 2. Recall ha J 0 () is a Bessel funcion of he rs kind and order zero. Theorem. Le e() = Then, for real posiive, ( ) k k 2 ln : 2k Z e() = 2 J 0 () d: We omi deails of he proof of Theorem. However, he reader should be able o consruc a proof by proceeding as in [4, I, p. 99], using he fac ha Z e J 0 (2) (6) d = 0: 0 A slighly more general resul han (6) is given in [2, equaion ] and is aribued o Nielsen [4]. An independen proof is given in [].

4 4 R. P. BRENT Corollary. Le e() be as in Theorem. Then, for large posiive, e() has an asympoic epansion e() = 2 =2 3=2 cos sin O 6 2 : For compuaional purposes, i is much beer o ake n = han n = 2 in (4), because he error for n = is O(e =). 4. A Dieren Generalizaion Equaion (4) may be obained from (2) by replacing k = by ( k =) n =n. An analogous generalizaion of (5) is (7) k n X = ln + + o() as!. Relaion (5) is jus he case n =. I is easy o show ha (7) is valid for all posiive ineger n. An essenial dierence beween (4) and (7) is ha here is a large amoun of cancellaion beween erms of size (e n (n+2)=2 ) on he lef side of (4), bu here is no cancellaion in he numeraor and denominaor on he lef side of (7). The funcion ( k =) n acs as a smoohing kernel wih a peak a k ' 2. In view of (), he resul (7) is no surprising, bu he speed of convergence may be surprising. Bren and McMillan [0] show ha (8) k n X = ln + + O(e cn ) ( if n =, as!, where c n = 2n sin 2 (=n) if n 2. In he case n = 2, he formula (8) has error O(e 4 ). Bren and McMillan [0] used his case wih ' 7,400 o compue o more han 30,000 decimal places. From Corollary, he same value of in (4) would give less han 8-decimal place accuracy. Also, more han 5,000 decimal places would have o be used in he compuaion o compensae for cancellaion of erms (e 2 = 2 ) in (4). The case n = 3 of (8) is ineresing because ma n=;2;::: c n = c 3 = 4:5: However, no one seems o have used n > 2 in a serious compuaion of. I would be ineresing o consider he behaviour of he funcions occurring in (4) and (8) for posiive bu non-inegral values of n. Cerainly (7) is valid for all posiive n, bu we do no know if (8) holds when n is posiive bu no an ineger (assuming a suiable eension of he deniion of c n ).

5 RAMANUJAN AND EULER'S CONSTANT 5 References. M. Abramowiz and I. A. Segun (eds.), Handbook of Mahemaical Funcions wih Formulas, Graphs, and Mahemaical Tables, Naional Bureau of Sandards, Washingon, 964 (reprined by Dover, 965). (Chaper was wrien by Y. L. Luke.) 2. R. Apery, Irraionalie de (2) e (3), Journees arihmeiques Luminy, Aserisque 6 (979), {3. 3. D. Bailey, Numerical resuls on he ranscendence of consans involving, e, and Euler's consan, Mah. Comp. 50 (988), 275{ B. C. Bernd, Ramanujan's Noebooks, Pars I{III, Springer-Verlag, New York, 985{ W. A. Beyer and M. S. Waerman, Error analysis of a compuaion of Euler's consan, Mah. Comp. 28 (974), 599{ J. M. Borwein and P. B. Borwein, Pi and he AGM, John Wiley and Sons, New York, J. M. Borwein, P. B. Borwein and D. H. Bailey, Ramanujan, modular equaions, and approimaions o pi or how o compue one billion digis of pi, Amer. Mah. Monhly 96 (989), 20{ R. P. Bren, Muliple-precision zero-nding mehods and he compleiy of elemenary funcion evaluaion, Analyic Compuaional Compleiy (J. F. Traub, ed.), Academic Press, New York, 975, 5{ R. P. Bren, Compuaion of he regular coninued fracion for Euler's consan, Mah. Comp. 3 (977), R. P. Bren and E. M. McMillan, Some new algorihms for high-precision compuaion of Euler's consan, Mah. Comp. 34 (980), 305{32. (There is an error on page 30: in he deniion of V p (z), \z=" should be \z k =".). R. P. Bren, An asympoic epansion inspired by Ramanujan, Ausral. Mah. Soc. Gaz. (o appear). Also Repor CMA-MR02-93, ANU, Feb. 993 (available by fp from dcssof.anu.edu.au in he direcory pub/bren). 2. I. S. Gradsheyn and I. M. Ryzhik, Tables of Inegrals, Series, and Producs, fourh ediion (rans. Alan Jerey), Academic Press, New York, Y. L. Luke, Inegrals of Bessel Funcions, McGraw-Hill, New York, N. Nielsen, Theorie des Inegrallogarihmus und verwander Transzendenen, Teubner, Leipzig, 906 (reprined by Chelsea, New York, 965). 5. J. Nunemacher, On compuing Euler's consan, Mah. Mag. 65 (992), 33{ F. W. J. Olver, Asympoics and Special Funcions, Academic Press, New York, A. van der Pooren, A proof ha Euler missed : : : Apery's proof of he irraionaliy of (3), Mah. Inelligencer (979), 95{ S. Ramanujan, A series for Euler's consan, Messenger of Mahemaics 46 (97), 73{80 (reprined in [9]). 9. S. Ramanujan, Colleced Papers of Srinivasa Ramanujan (G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson, eds.), Cambridge Universiy Press, Cambridge, 927. Reprined by Chelsea, New York, S. Ramanujan, Noebooks, wo volumes, Taa Insiue of Fundamenal Research, Bombay, E. Salamin, Compuaion of using arihmeic-geomeric mean, Mah. Comp. 30, 976, 565{ D. Sweeney, On he compuaion of Euler's consan, Mah. Comp. 7 (963), 70{78. Compuer Sciences Lab., Ausralian Naional Universiy, Canberra, ACT 0200, Ausralia address: rpb@cslab.anu.edu.au