RAMANUJAN AND PI JONATHAN M. BORWEIN

Transcription

1 RAMANUJAN AND PI JONATHAN M. BORWEIN Abstract. This cotributio highlights the progress made regardig Ramauja s work o Pi sice the ceteial of his birth i Ramauja ad Pi Sice Ramauja s 987 ceteial, much ew mathematics has bee stimulated by ucay formulas i Ramauja s Notebooks (lost ad foud). I illustratio, I metio the expositio by Moll ad his colleagues [] that illustrates various eat applicatios of Ramauja s Master Theorem, which extrapolates the Taylor coefficiets of a fuctio ad relates it to methods of itegratio used i particle physic]. I also ote lovely work o the modular fuctios behid Apéry ad Domb umbers by Cha ad others [6], ad fially I metio my ow work with Cradall o Ramauja s arithmetic-geometric cotiued fractio []. For reasos of space herei, I ow oly discuss work related directly to Pi ad so cotiue a story started i [9, ]. Truly ovel series for /π, based o elliptic itegrals, were foud by Ramauja aroud 90 [9, 5, 7, ]. Oe is: π = (4k)! ( k). (.) 980 (k!) k k=0 Each term of (.) adds eight correct digits. Though the uprove, Gosper used (.) for the computatio of a the-record 7 millio digits of π i 985 thereby completig the first proof of (.) [7, Ch. 3]. Soo after, David ad Gregory Chudovsky foud the followig variat, which relies o the quadratic umber field Q( 63) rather tha Date: May 3, 0. J. M. Borwei: Cetre for Computer Assisted Research Mathematics ad its Applicatios (CARMA), Uiversity of Newcastle, Callagha, NSW 308, Australia. joatha.borwei@ewcastle.edu.au.

2 JONATHAN M. BORWEIN Q( 58) as is implicit i (.): π = k=0 ( ) k (6k)! ( k) (3k)! (k!) k+3/. (.) Each term of (.) adds 4 correct digits. (Were a larger imagiary quadratic field to exist with class umber oe, there would be a eve more extravagat ratioal series for some surd divided by π [0].) The brothers used this formula several times, culmiatig i a 994 calculatio of π to over four billio decimal digits. Their remarkable story was told i a prizewiig New Yorker article [8]. Remarkably, (.) was used agai i 00 ad 0 for the curret record computatios of π to five ad te trillio decimal digits respectively... Quartic algorithm for π. The record for computatio of π has goe from 9.37 millio decimal digits i 986, to te trillio digits i 0. Sice the algorithm below which foud its ispiratio i Ramauja s 94 paper was used as part of computatios both the ad as late as as 009, it is iterestig to compare the performace i each case: Set a 0 := 6 4 ad y 0 :=, the iterate y k+ = ( y4 k )/4 + ( y 4 k )/4, a k+ = a k ( + y k+ ) 4 k+3 y k+ ( + y k+ + y k+). (.3) The a k coverges quartically to /π each iteratio quadruples the umber of correct digits. Twety-oe iteratios produce a algebraic umber that coicides with π to well more tha six trillio places. This scheme ad the 976 Salami Bret scheme [7, Ch. 3] have bee employed frequetly over the past quarter cetury. Here is a highly abbreviated chroology (based o wiki/chroology_of_computatio_of_pi). 986: David Bailey used (.3) to compute 9.4 millio digits of π. This required 8 hours o oe CPU of the ew Cray- at NASA Ames Research Ceter. Cofirmatio usig the Salami-Bret scheme took aother 40 hours. This computatio ucovered hardware ad software errors o the Cray-. Ja. 009: Takahashi used (.3) to compute.649 trillio digits (early 60,000 times the 986 computatio), requirig 73.5 hours o 04 cores (ad Tbyte memory) of a Appro Xtreme-X3 system. Cofirmatio via the Salami-Bret scheme took 64. hours ad 6.73 Tbyte of mai memory. Apr. 009: Takahashi computed.576 trillio digits.

3 RAMANUJAN AND PI 3 Dec. 009: Bellard computed early.7 trillio decimal digits (first i biary), usig (.). This took 3 days, but he oly used a sigle four-core workstatio with lots of disk storage ad eve more huma itelligece! Aug. 00: Kodo ad Yee computed 5 trillio decimal digits agai usig equatio (.). This was doe i biary, the coverted to decimal. The biary digits were cofirmed by computig 3 hexadecimal digits of π edig with positio 4,5,40,8,60, usig BBP-type formulas for π due to Bellard ad Plouffe [7, Chapter 3]. Additioal details are give at org/misc_rus/pi-5t/aouce_e.html. See also [4] i which aalysis, showig these digits appear to be very ormal, is made. Daiel Shaks, who i 96 computed π to over 00,000 digits, oce told Phil Davis that a billio-digit computatio would be forever impossible. But both Kaada ad the Chudovskys achieved that i 989. Similarly, the ituitioists Brouwer ad Heytig asserted the impossibility of ever kowig whether the sequece appears i the decimal expasio of π, yet it was foud i 997 by Kaada, begiig at positio As late as 989, Roger Perose vetured, i the first editio of his book The Emperor s New Mid, that we likely will ever kow if a strig of te cosecutive seves occurs i the decimal expasio of π. This strig was foud i 997 by Kaada, begiig at positio Figure shows the progress of π calculatios sice 970, superimposed with a lie that charts the log-term tred of Moore s Law. It is worth otig that whereas progress i computig π exceeded Moore s Law i the 990s, it has lagged a bit i the past decade. Most of this progress is still i mathematical debt to Ramauja. As oted, oe billio decimal digits were first computed i 989 ad the te (actually 50) billio digit mark was first passed i 997. Fiftee years later oe ca explore i real time multi-billio step walks o the hex digits of π at shtml as draw by Fra Arago... Formulas for /π ad more. About te years ago Jésus Guillera foud various Ramauja-like idetities for /π N, usig iteger relatio methods. The three most basic ad etirely ratioal idetities

4 4 JONATHAN M. BORWEIN are: Figure. Plot of π calculatios, i digits (dots), compared with the log-term slope of Moore s Law (lie). 4 π = π = 4 π 3? = ( ) + ( ) r() 5 ( ) (.4) 3 ( ) + ( ) r() 5 ( ) (.5) ( ) + r() 7 ( ), (.6) 8 where r() := (/ 3/ ( )/)/!. Guillera proved (.4) ad (.5) i tadem, by very igeiously usig the Wilf-Zeilberger algorithm [0, 7] for formally provig hypergeometric-like idetities [7, 5, ]. No other proof is kow. The third, (.6), is almost certaily true. Guillera ascribes (.6) to Gourevich, who foud it usig iteger relatio methods i 00. There are other sporadic ad uexplaied examples based o other symbols, most impressively a 00 discovery by Culle:? = π 4 ( 4 ) ( )7 ( 3 4 ) () 9 ( ) We shall revisit this formula below. ( ).3. Formulae for π. I 008 Guillera [5] produced aother lovely, if umerically iefficiet, pair of third-milleium idetities discovered (.7)

5 RAMANUJAN AND PI 5 with iteger relatio methods ad proved with creative telescopig this time for π rather tha its reciprocal. They are based o: ( x + 3 ( ) ) (6( + x) + ) = 8x, (.8) (x + ) 3 (x + ) ad ( x + 3 ) 6 (x + ) 3 (4( + x) + 5) = 3x ( x + ). (.9) (x + ) Here (a) = a(a + ) (a + ) is the risig factorial. Substitutig x = / i (.8) ad (.9), he obtaied respectively the formulae () 3 ( 3 ) 3 (3+) = π 4, () 3 6 ( 3 ) 3 (+3) = 4 π 3. (.0).4. Calabi-Yao equatios ad super-cogrueces. Motivated by the theory of Calabi-Yao differetial equatios [], Almkvist ad Guillera have discovered may ew idetities. Oe of the most pleasig is: =? 3 π 3 (6 )! ( ) (!) 6. (.) This is yet oe more case where mysterious coectios have bee foud betwee disparate parts of mathematics ad Ramauja s work [, 3, 4]. As a fial example, we metio the existece of super-cogrueces of the type described i [3, 6, 3]. These are based o the empirical observatio that a Ramauja series for /π N, if trucated after p terms for a prime p, seems always to produce cogrueces to a higher power of p. The formulas below are take from []: p ( ) 4 ( )3 ( 3) 4 ( ) p ( mod p 5 ) (.) 4 () 5 p ( 4 ) ( )7 ( 3 4 ) ( )? p 4 ( mod p 9 ). () 9 We ote that (.3) is the super-cogruece correspodig to (.6), while for (.) the correspodig ifiite series sums to 3/π 4. We coclude by remidig the reader that all idetities marked with? = are assuredly true but remai to be prove. Ramauja might well be pleased. (.3)

6 6 JONATHAN M. BORWEIN Refereces [] T. Amdeberha, O. Espiosa, I. Gozalez, M. Harriso, V. Moll, ad A. Straub. Ramauja s Master Theorem. Ramauja J., i Press, 0. [] G. Almkvist. Ramauja-like formulas for /π á la Guillera ad Zudili ad Calabi-Yau differetial equatios. Computer Sciece Joural of Moldova, 7, o. (49), (009), [3] G. Almkvist ad A. Meurma. Jesus Guillera s formula for /π ad supercogrueces. (Swedish) Normat 58 o. (00), [4] D.H. Bailey, J.M. Borwei, C. S. Calude, M. J. Diee, M. Dumitrescu, ad A. Yee. A empirical approach to the ormality of pi. Experimetal Mathematics. Accepted February 0. [5] N.D. Baruah, B.C. Berdt, ad H.H. Cha. Ramauja s series for /π: A survey. Amer. Math. Mothly 6 (009), [6] B.C. Berdt, H.H. Cha ad S.S. Huag. Icomplete Elliptic Itegrals i Ramauja s Lost Notebook. Pp i q-series from a cotemporary perspective. Cotemp. Math., 54, Amer. Math. Soc., Providece, RI, 000. [7] J.M. Borwei ad D.H. Bailey. Mathematics by Experimet: Plausible Reasoig i the st Cetury. Ed, A K Peters, Natick, MA, 008. [8] J.M. Borwei, D.H. Bailey ad R. Girgesoh. Experimetatio i Mathematics: Computatioal Roads to Discovery. A K Peters, 004. [9] J.M. Borwei ad P.B. Borwei. Ramauja ad Pi. Scietific America, February 988, 7. Reprited i pp of Ramauja: Essays ad Surveys, B.C. Berdt ad R.A. Raki Eds., AMS-LMS History of Mathematics, vol., 00. [0] J.M. Borwei ad P.B. Borwei. Class umber three Ramauja type series for /π. Joural of Computatioal ad Applied Math (Special Issue), 46 (993), [] J.M. Borwei, P.B. Borwei, ad D.A. Bailey. Ramauja, modular equatios ad pi or how to compute a billio digits of pi. MAA Mothly, 96 (989), 0 9. Reprited i [] J.M. Borwei ad R.E. Cradall. O the Ramauja AGM fractio. Part II: the Complex-parameter Case. Experimetal Mathematics, 3 (004), [3] H.H. Cha, J. Wa ad W. Zudili. Legedre polyomials ad Ramauja-type series for /π. Israel J. Math.. I presss, 0. [4] H.H. Cha, J. Wa ad W. Zudili. Complex series for /π. Ramauja J. I press, 0. [5] J. Guillera. Hypergeometric idetities for 0 exteded Ramauja-type series. Ramauja Joural, vol. 5 (008), pp [6] J. Guillera ad W. Zudili. Diverget Ramauja-type supercogrueces. Proc. Amer. Math. Soc. 40 (0), [7] M. Petkovsek, H. S. Wilf, D. Zeilberger. A = B. A K Peters, 996. [8] R. Presto. (99) The Moutais of Pi. New Yorker, Mar 99, http: // [9] S. Ramauja, Modular equatios ad approximatios to Pi. Quart. J. Pure. Appl. Math. 45 (93-94), [0] H. S. Wilf ad D. Zeilberger, Ratioal Fuctios Certify Combiatorial Idetities. Joural of the AMS, 3 (990),

7 RAMANUJAN AND PI 7 [] W. Zudili. Ramauja-type formulae for /π: a secod wid? Modular forms ad strig duality. Pages i Fields Ist. Commu., 5, Amer. Math. Soc., Providece, RI, 008. [] W. Zudili. Arithmetic hypergeometric series. Russia Math. Surveys 66: (0), 5. [3] W. Zudili. Ramauja-type supercogrueces. J. Number Theory 9 (009),