Charles Vanden Eynden

Transcription

1 RAMANUJAN S SERIES FOR /π: A SURVEY NAYANDEEP DEKA BARUAH, BRUCE C. BERNDT 2, and HENG HUAT CHAN 3 { To Charles Vanden Eynden on the occasion of his } /4 birthday Abstract. Beginning with Ramanuan s epic paper, Modular Equations and Approximations to π, we describe Ramanuan s series for /π and later attempts to prove them. Generalizations, analogues, and consequences of Ramanuan s series are discussed.. The Beginning Toward the end of the first paper [55], [56, p. 36] that Ramanuan published in England, at the beginning of Section 3, he writes, I shall conclude this paper by giving a few series for /π. In fact, Ramanuan concluded his paper a couple pages later with another topic: formulas and approximations for the perimeter of an ellipse.) After sketching his ideas, which we examine in detail in Sections 3 and 9, Ramanuan records three series representations for /π. As customary, set Let a) 0 :=, a) n := aa + ) a + n ), n. A n := 2 )3 n, n 0..) n! 3 Research partially supported by BOYSCAST Fellowship grant SR/BY/M-03/05 from DST, Govt. of India. 2 Research partially supported by grant H from the National Security Agency. 3 Research partially supported by National University of Singapore Academic Research Fund R

2 2 Baruah, Berndt, and Chan Theorem.. If A n is defined by.), then 4 π = 6n + )A n 4,.2) n n=0 6 π = 42n + 5)A n,.3) 26n n=0 32 π = )n + 5 ) ) 8n 5 5 A n..4) 2 6n 2 n=0 The first two formulas,.2) and.3), appeared in the Walt Disney film, High School Musical, starring Vanessa Anne Hudgens, who plays an exceptionally bright high school student named Gabriella Montez. Gabriella points out to her teacher that she had incorrectly written the left-hand side of.3) as 8/π instead of 6/π on the blackboard. After first claiming that Gabriella is wrong, her teacher checks possibly Ramanuan s Collected Papers?) and admits that Gabriella is correct. Formula.2) was correctly recorded on the blackboard. After offering the three formulas for /π given above, at the beginning of Section 4 [55], [56, p. 37], Ramanuan claims, There are corresponding theories in which q is replaced by one or other of the functions q r := q r x) := exp π cscπ/r) 2 F, r ; ; x) r r 2F, r ; ; x) r r ),.5) where r = 3, 4, or 6, and where 2 F denotes one of the hypergeometric functions p F p, p, which are defined by a ) n a p ) n x n pf p a,..., a p ; b,..., b p ; x) :=, x <. b ) n b p ) n n! Ramanuan then offers 4 further series representations for /π. Of these, 0 belong to the quartic theory, i.e., for r = 4; 2 belong to the cubic theory, i.e., for r = 3; and 2 belong to the sextic theory, i.e., for r = 6. Ramanuan never returned to the corresponding theories in his published papers, but six pages in his second notebook [57] are devoted to developing these theories, with all of the results on these six pages being proved in a paper [5] by Berndt, S. Bhargava, and F. G. Garvan. That the classical hypergeometric function 2 F, ; ; x) in the classical theory of elliptic functions 2 2 could be replaced by one of the three hypergeometric functions above and concomitant theories developed is one of the many incredibly ingenious and useful ideas bequeathed to us by Ramanuan. The development of these theories is far from easy and is an active area of contemporary research. All 7 series for /π were discovered by Ramanuan in India before arriving in England, for they can be found in his notebooks [57], which were written prior to Ramanuan s departure for England. In particular,.2),.3), and.4) can be found on page 355 in his second notebook and the remaining 4 series are found in his third notebook [57, p. 378]; see also [3, pp ]. It is interesting that.2),.3), and.4) are n=0

3 Series for /π 3 also located on a page published with Ramanuan s lost notebook [58, p. 370]; see also [3, Chapter 5]. 2. The Main Actors Following in the Footsteps of Ramanuan Fourteen years after the publication of [55], the first mathematician to address Ramanuan s formulas was Sarvadaman Chowla [35], [36], [37, pp. 87 9, 6 9], who gave the first published proof of a general series representation for /π and used it to derive the first.2) of Ramanuan s series for /π [55, Eq. 28)]. We briefly discuss Chowla s ideas in Section 4. Ramanuan s series were then forgotten by the mathematical community until November, 985, when R. William Gosper, Jr. used one of Ramanuan s series, namely, 980 π 8 = n=0 4n)! n) n!) n, 2.) to calculate 7,526,00 digits of π, which at that time was a world record. There was only one problem with his calculation 2.) had not yet been proved. However, a comparison of Gosper s calculation of the digits of π with the previous world record held by Y. Kanada made it extremely unlikely that 2.) was incorrect. In 987, Jonathan and Peter Borwein [22] succeeded in proving all 7 of Ramanuan s series for /π. In a subsequent series of papers [23], [24], [28], they established several further series for /π, with one of their series [28] yielding roughly fifty digits of π per term. The Borweins were also keen on calculating the digits of π, and accounts of their work can be found in [29], [27], and [25]. At about the same time as the Borweins were devising their proofs, David and Gregory Chudnovsky [38] also derived series representations for /π and, in particular, used their series π = 2 ) n 6n)! n 2.2) n!) 3 3n)! ) 3n+3/2 n=0 to calculate a world record 2,260,33,336 digits of π. The series 2.2) yields 4 digits of π per term. A popular account of the Chudnovskys calculations can be found in a paper written for The New Yorker [54]. The third author of the present paper and his coauthors Berndt, S. H. Chan, A. Gee, W. C. Liaw, Z. G. Liu, V. Tan, and H. Verrill) in a series of papers [8], [30], [3], [32], [34] extended the ideas of the Borweins, in particular, without using Clausen s formula in [30] and [34], and derived general hypergeometric-like formulas for /π. We devote Section 8 of our survey to discussing some of their results. Stimulated by the work and suggestions of the third author, the first two authors [7], [8] systematically returned to Ramanuan s development in [55] and employed his ideas in order not only to prove most of Ramanuan s original representations for /π but also to establish a plethora of new such identities as well. In another paper [9], motivated by the work of Jesús Guillera [46] [5], who both experimentally and rigorously discovered many new series for both /π and /π 2, the first two authors continued to follow Ramanuan s ideas and devised series representations for /π 2.

4 4 Baruah, Berndt, and Chan In the survey which follows, we delineate the main ideas in Sections 3, 6, 7, 8, and 9, where the ideas of Ramanuan, the Borwein brothers, the Chudnovsky brothers, Chan and his coauthors, and the present authors, respectively, are discussed. 3. Ramanuan s Ideas To describe Ramanuan s ideas, we need several definitions from the classical theory of elliptic functions, which, in fact, we use throughout the paper. The complete elliptic integral of the first kind is defined by K := Kk) := π/2 0 dϕ k2 sin 2 ϕ, 3.) where k, 0 < k <, denotes the modulus. Furthermore, K := Kk ), where k := k2 is the complementary modulus. The complete elliptic integral of the second kind is defined by E := Ek) := π/2 0 k 2 sin 2 ϕ dϕ. If q = exp πk /K), then one of the central theorems in the theory of elliptic functions asserts that [2, p. 0, Entry 6] ϕ 2 q) = 2 π Kk) = 2F 2, 2 ; ; k2 ), 3.2) where ϕq) in Ramanuan s notation or ϑ 3 q) in the classical notation) denotes the classical theta function defined by ϕq) = q ) = Note that, in the notation.5), q = q 2. The second equality in 3.2) follows from expanding the integrand in a binomial series and integrating termwise. Let K, K, L, and L denote complete elliptic integrals of the first kind associated with the moduli k, k, l, and l, respectively. Suppose that, for some positive integer n, n K K = L L. 3.4) A modular equation of degree n is an equation involving k and l that is induced by 3.4). Modular equations are always algebraic equations. An example of a modular equation of degree 7 may be found later in 9.8). Alternatively, by 3.2), 3.4) can be expressed in terms of hypergeometric functions. We often say that l has degree n over k. Derivations of modular equations ultimately rest on 3.2). If we set K/K = n, so that q = e πn, then kq) is called the singular modulus. The multiplier m = mq) is defined by m := mq) := 2 F 2, 2 ; ; k2 q) ) 2F 2, 2 ; ; l2 q) ). 3.5)

5 Series for /π 5 We note here that k 2 q) and mq) can be represented by [2, Entry 3, p. 98; Entry 24ii), p. 24] k 2 q) = 6q ψ4 q 2 ) and mq) = ϕ2 q) ϕ 4 q) ϕ 2 q n ), respectively, where ϕq) is defined by 3.3) and ψq) = q +)/2. Thus, modular equations can also be written as theta function identities. Ramanuan begins Section 3 of [55] with a special case of Clausen s formula [22, p. 78, Proposition 5.6b)], 4K 2 π 2 = 2 )3!) 3 2kk ) 2 = 3 F 2 2, 2, 2 ;, ; 2kk ) 2), 3.6) which can be found as Entry 3 of Chapter in his second notebook [57] [, p. 58]. Except for economizing notation, we now quote Ramanuan. Hence we have ) 2/3 q /3 q 2 ; q 2 ) 4 = 4 kk 2 )3!) 3 2kk ) 2, 3.7) where a; q) := a) aq) aq 2 ). 3.8) Differentiating both sides in 3.7) logarithmically with respect to k, we can easily shew that q 2 24 q = 2 2k2 ) 3 + ) 2 )3!) 3 2kk ) ) But it follows from = 3 π n 24 = e 2π n = ) 2 K Ak) 3.0) π where Ak) is a certain type of algebraic number, that, when q = e πn, n being a rational number, the left-hand side of 3.9) can be expressed in the form ) 2 2K A + B π π, where A and B are algebraic numbers expressible by surds. Combining 3.6) and 3.9) in such a way as to eliminate the term 2K/π) 2, we are left with a series for /π. He then gives the three examples.2).4). Ramanuan s ideas will be described in more detail in Section 9. However, in closing this section, we note that the series on the left-hand sides of 3.9) and 3.0) is Ramanuan s Eisenstein series P q 2 ), with q = e πn in the latter instance, where P q) := 24 = q, q <. 3.) q

6 6 Baruah, Berndt, and Chan Ramanuan s derivation of 3.0) arises firstly from the transformation formula for P q), which in turn is an easy consequence of the transformation formula for the Dedekind eta-function, given in 9.) below. The second ingredient in deriving 3.0) is an identity for np q 2n ) P q 2 ) in terms of the moduli k and l, where l has degree n over k. Formula 3.7) follows from a standard theorem in elliptic functions that Ramanuan also recorded in his notebooks [57], [2, p. 24, Entry 2iii)]. 4. Sarvadaman Chowla Chowla s ideas reside in the classical theory of elliptic functions and are not unlike those that the Borweins employed several years later. We now briefly describe Chowla s [35], [36], [37, pp. 87 9, 6 9] approach. Using classical formulas of Cayley and Legendre relating the complete elliptic integrals K and E, he specializes them by setting K/K = n. He then defines and S r := T r := r 2 )3!) 3 2kk ) 2 4.) r 4 )2!) 2 2kk ) ) Chowla then writes Then it is known that, when k / 2, 2K π = + T 0 and 4K 2 π 2 = + S ) Chowla does not give his source for either formula, but the second formula in 4.3), as noted above, is a special case of Clausen s formula 3.6). The first formula is a special case of Kummer s quadratic transformation [22, pp ], which was also known to Ramanuan. Each of the formulas of 4.3) is differentiated twice with respect to k, and, without giving details, Chowla concludes that if K/K = n, then K = a + b T 0 + c T, K π = d T + e T 2, π = a 2S 0 + b 2 S, K 2 = a 3 + b 3 S 0 + c 3 S + d 3 S 2, where a, b,... are algebraic numbers. He then sets n = 3 and k = sinπ/2) in each of the four formulas above to deduce, in particular, identity.2) from the second formula above.

7 Series for /π 7 5. R. William Gosper, Jr. As we indicated in the Introduction, in November, 985, Gosper employed a lisp machine at Symbolics and Ramanuan s series 2.) to calculate 7,526,00 digits of π, which at that time was a world record. Of the 7 series found by Ramanuan, this one converges the fastest, giving about 8 digits of π per term. At the time of Gosper s calculation, the world record for digits of π was about 6 million digits calculated by Y. Kanada. Before the Borwein brothers had later found a conventional proof of Ramanuan s series 2.), they had shown that either 2.) yields an exact formula for π or that it differs from π by more than Thus, by demonstrating that his calculation of π agreed with that of Kanada, Gosper effectively had completed the Borweins first proof of 2.). However, Gosper s primary goal was not to eclipse Kanada s record but to study the simple) continued fraction expansion of π for which he calculated 7,00,303 terms. In letters [45] from February, 992 and May, 993, Gosper offered the following remarks on his calculations: Of course, what the scribblers always censor is that the digits were a by-product. I wanted to change the obect of the game away from meaningless decimal digits. I used what I call a resumable matrix tower to exactly compute an enormous rational equal to the sum of a couple of million terms of Ramanuan s 99 4n series. I then divided to form the binary integer = floorπ2 58,000,000 ). I converted this to decimal and sent a summary to Kanada for comparison, and converted the binary fraction to a cf using an fft based scheme. Gosper s world record was short lived, as in January, 986, D. H. Bailey used an algorithm of the Borweins arising from a fourth order modular equation to compute 29,360,000 digits of π. Gosper s calculation of the continued fraction expansion of π was motivated by the fact that many important mathematical constants do not have interesting decimal expansions but do have interesting continued fraction expansions. That continued fraction expansions are considerably more interesting than decimal expansions is a view shared by the Chudnovsky brothers [4]. Continued fraction expansions can often be used to distinguish a constant from others, while decimal expansions likely will be unable to do so. For example, the simple continued fraction of e, namely, e = has a pattern. On the contrary, taking a large random string of digits of e would not help one identify e. It is an open problem if the simple continued fraction of π, namely, π = has a pattern

8 8 Baruah, Berndt, and Chan Gosper also derived a hypergeometric-like series representation for π, namely, 50 6 π = ), 5.) 2 which can be used to calculate any particular binary digit of π. See a paper by G. Almkvist, C. Krattenthaler, and J. Petersson [] for a proof of 5.) as well as generalizations, which include the following theorem. Theorem 5.. For each integer k, there exists a formula S k ) π = ), 4) k 8k 4k where S k ) is a polynomial in of degree 4k with rational coefficients Jonathan and Peter Borwein One key to the work of both Ramanuan and Chowla in their derivations of formulas for /π is Clausen s formula for the square of a complete elliptic integral of the first kind or, by 3.2), for the square of the hypergeometric function 2 F, ; ; 2 2 k2 ). The aforementioned rendition 3.6) of Clausen s formula is not the most general version of Clausen s formula, namely, 2F 2 a, b; a + b + ; z) = 2 3 F 2 2a, 2b, a + b; a + b +, 2a + 2b; z). 6.) 2 Indeed, the work of many authors who have proved Ramanuan-like series for /π ultimately rests on special cases of 6.). In particular, squares of certain other hypergeometric functions lead one to Ramanuan s alternative theories of elliptic functions. A second key step is to find another formula for K/π) 2, which also contains another term involving /π. Combining the two formulas to eliminate the term K/π) 2 then produces a hypergeometric-type series representation for /π. Evidently, unaware of Chowla s earlier work, the Borweins proceeded in a similar fashion and used Legendre s relation [22, p. 24] Ek)K k) + E k)kk) Kk)K k) = π 2 and other relations between elliptic integrals to produce such formulas. Having derived a series representation for /π, one now faces the problem of evaluating the moduli and elliptic integrals that appear in the formulas. When q = e πn, then for certain positive integers n one can evaluate the requisite quantities. This leads us to the definition of the Ramanuan Weber class invariants. After Ramanuan, set χq) := q; q 2 ), q <. 6.2) If n is a positive rational number and q = e πn, then the class invariants G n and g n are defined by G n := 2 /4 q /24 χq) and g n := 2 /4 q /24 χ q). 6.3) In the notation of H. Weber [62], G n = 2 /4 f n) and g n = 2 /4 f n). As mentioned in Section 3, k n := ke πn ) is called the singular modulus. In his voluminous

9 Series for /π 9 work on modular equations, Ramanuan sets α := k 2 and β := l 2. Accordingly, we set α n := k 2 n. Because [2, p. 24, Entries 2v), vi)] χq) = 2 /6 {α α)/q} /24 and χ q) = 2 /6 {α α) 2 /q} /24, it follows from 6.3) that G n = {4α n α n )} /24 and g n = {4α n α n ) 2 } / ) Noting the arguments on the right-hand side of 3.6) when q = e πn, we see the importance of 6.4). The singular moduli and class invariants are actually algebraic numbers. In general, as n increases, the corresponding series for /π converges more rapidly. The series 2.2) is associated with the imaginary quadratic field Q 63). The Borweins proofs of all 7 of Ramanuan s series for /π can be found in their book [22]. Their derivations arise from several general hypergeometric-like series representations for /π given in terms of singular moduli, class invariants, and complete elliptic integrals [22, pp. 8 84]. Another account of their work, but with fewer details, can be found in their paper [23] commemorating the centenary of Ramanuan s birth. Further celebrating the 00th anniversary of Ramanuan s birth, the Borweins derived further series for /π in [24]. The series in this paper correspond to imaginary quadratic fields with class number 2, with one of their series corresponding to d = 427 and yielding about 25 digits of π per term. In [28], the authors derived series for /π arising from fields with class number 3, with a series corresponding to d = 907 yielding about 37 or 38 digits of π per term. Their record is a series associated with a field of class number 4 giving about 50 digits of π per term; here, d = 555 [27]. The latter paper gives the details of what we have written in this paragraph. The Borweins have done an excellent ob of communicating their work to a wide audience. Besides their paper [27], see their paper in the American Mathematical Monthly [29], with D. H. Bailey, on computing π, especially via work of Ramanuan, and their delightful paper in the Scientific American [25], which has been reprinted in [2, pp ]. 7. David and Gregory Chudnovsky In our Introduction we mentioned that the Chudnovsky brothers, Gregory and David, used 2.2) to calculate over 2 billion digits of π. They had first used 2.2) to calculate,30,60,664 digits of π in the fall of 989 on a borrowed computer. They then built their own computer, m zero, described colorfully in [54], and set a world record of 2,260,32,336 digits of π. The world record for digits of π has been broken several times since then, and since it is not the purpose of this paper to delineate this computational history, we refrain from mentioning further records. The Chudnovskys, among others, have extensively examined their calculations for patterns. It is a long outstanding conecture that π is normal. In particular, if N denotes the number of calculated digits of π, then for each k, 0 k 9, # of appearances of k in the first N digits lim N N = 0.

10 0 Baruah, Berndt, and Chan The Chudnovskys calculations, and all subsequent calculations of Kanada, lend credence to this conecture. As a consequence, the average of the digits over a long interval should be approximately 4.5. The Chudnovskys found that for the first billion digits the average stays a bit on the high side, while for the next billion digits, the average hovers a bit on the low side. Their paper [4] gives an interesting statistical analysis of the digits up to one billion. For example, strings of consecutive digits of maximal lengths between 8 and 0 occur for each digit. The Chudnovskys deduced 2.2) from a general series representation for /π for which we need to make several definitions in order to describe. For τ H = {τ : Im τ > 0} and each positive integer k, the Eisenstein series E 2k τ) is defined by E 2k τ) := 4k B 2k σ 2k )q, q = e πiτ, 7.) = where B k, k 0, denotes the kth Bernoulli number and σ k n) = d n dk. Klein s absolute modular J-invariant is defined by Jτ) := E4τ) 3, τ H. 7.2) E4τ) 3 E6τ) 2 It is well known that if αq) := k 2 q), where k is the modulus, then [2, pp , Entry 3] J2τ) = 4 αq) + α2 q)) 3 27α 2 q) αq)) ) Thus, 6.4) and 7.3) show that, when q = e πn, singular moduli, class invariants, and the modular J-invariant are intimately related. Now define s 2 τ) := E 4τ) E 2 τ) 3 ). E 6 τ) π Im τ We are now ready to state the Chudnovskys main formula [42, p. 22]. If τ = + d)/2, then { } 6µ)! 6 s Jτ) 2τ)) + µ 3µ)!µ! µ J µ τ) =. 7.4) π d Jτ)) µ=0 The Chudnovskys series 2.2) is the special case d = 63 of 7.4). The Chudnovsky brothers developed and extended Ramanuan s ideas in directions different from those of other authors. They obtained hypergeometric-like representations for other transcendental constants and proved, for example, that Γ ) Γ ) and π Γ 3) Γ ) 4 are transcendental [40]. They extended their analysis to linear forms of logarithms. Their advances involve the second solution of the hypergeometric differential equation. Recall from the theory of linear differential equations that 2 F a, b; c; x) is a solution of a certain second order linear differential equation with a regular singular

11 Series for /π point at the origin [5, p. ]. A second linearly independent solution is generally not analytic at the origin, and in [4] and [42, pp ], the Chudnovsky brothers establish new hypergeometric series identities involving the latter function. Their identities in this paper lead to hypergeometric-like series representations for π, including Gosper s formula 5.). In [43], the authors provide a lengthy list of such examples, including 45π = ) ). 4 The Chudnovsky brothers have also employed series for /π to derive theorems on transcendence measures µα), which are defined by { µα) := inf µ > 0 : 0 < α p q < q has finitely many solutions p } µ q Q. By the famous Thue Siegel Roth Theorem [6, p. 66], =, if α is rational, µα) = 2, if α is algebraic but not rational, 2, if α is transcendental. Although the Chudnovskys can obtain irrationality measures for various constants, the one they obtain for π is not as good as one would like. Currently, the world record for the irrationality measure of π is held by M. Hata [52], who proved that µπ) Their methods are much better in obtaining irrationality measures for expressions such as π/ , arising in their series 2.2). See also a paper by W. Zudilin [64]. 8. Ramanuan s Cubic Class Invariant and His Alternative Theories In Sections 3, 4, 6, and 7, we emphasized how Ramanuan Weber class invariants and singular moduli were of central importance for Ramanuan and others who followed in deriving series for /π. We also stressed in Section, in particular, in the discourse after.5), that Ramanuan s remarkable observation of replacing the classical hypergeometric function 2 F, ; ; x) by 2 2 2F, r ; ; x), r = 3, 4, 6, leads to new r r and beautiful alternative theories. On the top of the page 22 in his lost notebook [58], Ramanuan defines a cubic class invariant λ n i.e., r = 3 in.5)), which is an analogue of the Ramanuan Weber classical invariants G n and g n defined in 6.3). Define Ramanuan s function f q) := q; q), q <, 8.) where a; q) is defined in 3.8), and the Dedekind eta-function ητ) ητ) := e 2πiτ/24 e 2πiτ ) =: q /24 f q), 8.2) =

12 2 Baruah, Berndt, and Chan where q = e 2πiτ and Im τ > 0. Then Ramanuan s cubic class invariant λ n is defined by ) 6 + i n/3 λ n = 3 f 6 q) 3 qf 6 q 3 ) = η 3 2 ), 8.3) 3 + i 3n η 2 where q = e πn/3, i.e., τ = i n/3. 2 Chan, Liaw, and Tan [32] established a general series representation for /π in terms of λ n that is analogous to the general formulas of the Borweins and Chudnovskys in terms of the classical class invariants. To state this general formula, we first need some definitions. Define α q) := 27q f 2 q) ) f 2 q 3 ) Thus, when q = e πn/3 and αn := α e πn/3 ), 8.3) and 8.4) imply that = λ 2 n. In analogy with 3.5), define the multiplier mq) by α n mq) := mα, β ) := 2 F, 2 ; ; α ) 3 3 2F, 2 ; ; β ), 3 3 where β = α q n ). We are now ready to state the general representation of /π derived by Chan, Liaw, and Tan [32, p. 02, Theorem 4.2]. Theorem 8.. For n, let a n = α n αn) dmα, β ) n dα α = α n,β =αn, and Then b n = 2α n, H n = 4α n α n). 3 π n = 2 2) 3) 3) a n + b n ) H!) n. 8.5) 3 We give one example. Let n = 9; then α9 = 9. Then, without providing further 8 2 2) 3) 3) details, 4 π 3 = 5 + )!) 3 9 6), which was discovered by Chan, Liaw, and Tan [32, p. 95]. Another general series representation for /π in the alternative theories of Ramanuan was devised by Berndt and Chan [8, p. 88, Eq. 5.80)]. We will not state this

13 Series for /π 3 formula and all the requisite definitions, but let it suffice to say that the formula involves Ramanuan s Eisenstein series P q), Qq), and Rq) at the argument q = e πn and the modular -invariant, defined by τ) = 728Jτ), where Jτ) is defined by 7.2). In particular, ) 3 + 3n = 27 λ2 n )9λ 2 n ) 3, 2 λ 2 n a proof of which can be found in [7]. The hypergeometric terms are of the form 5 2) 6) 6)!) 3. Berndt and Chan used their general formula to calculate a series for /π that yields about 73 or 74 digits of π per term. Lastly, we conclude this section by remarking that Berndt, Chan, and Liaw [9] have derived series representations for /π that fall under the umbrella of Ramanuan s quartic theory of elliptic functions. Because the quartic theory is intimately connected with the classical theory, their general formulas [9, p. 44, Theorem 4.] involve the classical invariants G n and g n in their summands. Not surprisingly, the hypergeometric terms are of the form 3 2) 4) 4) B :=.!) 3 The simplest example arising from their theory is given by [9, p. 45] 9 2π = ) 32 B 7 + ) The Present Authors As Disciples of Ramanuan As mentioned in Section 2, the first two authors were inspired by the third author to continue the development of Ramanuan s thoughts. In their first paper [7], Baruah and Berndt employed Ramanuan s ideas in the classical theory of elliptic functions to prove 3 of Ramanuan s original formulas and many new ones as well. In [8], they utilized Ramanuan s cubic and quartic theories to establish five of Ramanuan s 7 formulas in addition to some new representations. Lastly, in [9], motivated by the work of J. Guillera, described briefly in Section 0 below, the first two authors extended Ramanuan s ideas to derive hypergeometric-like series representations for /π 2. For example, 24 π = ) µ µ µ )B 2 µ, 99 4 where µ=0 B µ = µ 4) ν=0 ν 4) µ ν 2) ν 2) µ ν 3 4) ν 3 4) µ ν ν! 3 µ ν)! 3. In Section 3, we defined Ramanuan s Eisenstein series P q) in 3.) and offered several definitions from Ramanuan s theories of elliptic functions in giving a brief

14 4 Baruah, Berndt, and Chan introduction to Ramanuan s ideas. Here we highlight the role of P q) in more detail before giving a complete proof of.3). Because these three series representations.2).4) can also be found in Ramanuan s lost notebook, our proof here is similar to that given in [3, Chapter 5]. Following Ramanuan, set z := 2 F 2, 2 ; ; x). 9.) The two most important ingredients in our derivations are Ramanuan s representation for P q 2 ) given by [2, p. 20, Entry 9iv)] P q 2 ) = 2x)z 2 + 6x x)z dz dx and Clausen s formula 3.6), which, using 9.), we restate in the form where, as in.), From 9.3) and 9.4), 9.2) z 2 = 3 F 2,, ;, ; X) = A X, 9.3) A := 2 )3! 3 and X := 4x x). 9.4) 2z dz dx = A X 4 2x). 9.5) Hence, from 9.2), 9.3), 9.5), and 9.4), P q 2 ) = 2x) A X + 3 2x) A X = { 2x) + 3 2x)}A X. 9.6) For q := e πn, recall 9.) and set x n = k 2 e πn ), z n := 2 F, ; ; x ) 2 2 n 9.7) and X n = 4x n x n ). 9.8) For later use, we note that [3, p. 375] x n = x /n and z /n = nz n. 9.9) With the use of 9.7) and 9.8), 9.6) takes the form P e 2πn ) = { 2x n ) + 3 2x n )}A Xn = 2x n )z 2 n + 3 2x n )A Xn. 9.0)

15 Series for /π 5 In order to utilize 9.0), we require two different formulas, each involving both P q 2 ) and P q 2n ), where n is a positive integer. The first comes from a transformation formula for P q), which in turn arises from the transformation formula for f q) defined in 8.) or the Dedekind eta function defined in 8.2), and is for general n. This transformation formula is given by [2, p. 43, Entry 27iii)] e α/2 α /4 f e 2α ) = e β/2 β /4 f e 2β ), 9.) where αβ = π 2, with α and β both positive. Taking the logarithm of both sides of 9.), we find that α log α + log e 2α ) = β log β + log e 2β ). 9.2) = Differentiating both sides of 9.2) with respect to α, we deduce that 2 + 4α + = = 2e 2α e = β 2α 2α 4α = = 2β/α)e 2β e 2β. 9.3) Multiplying both sides of 9.3) by 2α and rearranging, we arrive at ) ) e 2α e 2β 6 α 24 = β ) e 2α e 2β Setting α = π/ n, so that β = π n, recalling the definition 3.) of P q), and rearranging slightly, we see that 9.4) takes the shape 6 n π = = P n e 2π/ ) + np e 2πn ). 9.5) This is the first desired formula. The second gives representations for Ramanuan s function [55], [56, pp ] f n q) := np q 2n ) P q 2 ) 9.6) for certain positive integers n. Ramanuan [55], [56, pp ] used the notation fn) instead of f n q).) In [55], Ramanuan recorded representations for f n q) for 2 values of n, but he gave no indication of how these might be proved. These formulas are also recorded in Chapter 2 of Ramanuan s second notebook [57], and proofs may be found in [2]. We now give the details for our proof of.3), which was clearly a favorite of Gabriella Montez, the precocious student in High School Musical. Unfortunately, we do not know whether she possessed a proof of her own. We restate.3) here for convenience. Theorem 9.. If A, 0, is defined by 9.4), then 6 π = )A. 9.7) 26

16 6 Baruah, Berndt, and Chan Proof. The identity 9.7) is connected with modular equations of degree 7. Thus, our first task is to calculate the singular modulus x 7. To that end, we begin with a modular equation of degree 7 { xq)xq 7 ) } /8 + { xq)) xq 7 )) } /8 =, 9.8) due to C. Guetzlaff in 834 but rediscovered by Ramanuan in Entry 9i) of Chapter 9 of his second notebook [57], [2, p. 34]. In the notation of our definition of a modular equation after 3.4), we have set xq) = k 2 q), and so xq 7 ) = l 2 q). Set q = e π/7 in 9.8) and use 9.9) and 9.8) to deduce that 2 {x 7 x 7 )} /8 = and X 7 = ) Ramanuan calculated the singular modulus x 7 in his first notebook [57], [4, p. 290], from which, or from 9.9), we easily can deduce that 2x 7 = ) In the notation 9.6), from either [55], [56, p. 33] or [2, p. 468, Entry 5iii)], f 7 q) = 3zq)zq 7 ) + xq)xq 7 ) + ) xq)) xq 7 )). 9.2) Putting q = e π/7 in 9.2) and employing 9.9) and 9.9), we find that f 7 e π/7 ) = ) x 7 x 7 ) z7 2 = z ) Letting n = 7 in 9.5) and 9.0), and using 9.20), we see that and 6 7 π = P 7 e 2π/ ) + 7P e 2π7 ) 9.23) P e 2π7 ) = 2x 7 )z x 7 )A X 7 = z A, 9.24) 26 respectively. Eliminating P e 2π/7 ) from 9.22) and 9.23) and putting the resulting formula for P e 2π7 ) in 9.24), we find that 3 7 7π z2 7 = z A 8 2, 6 which upon simplification with the use of 9.3) yields 9.7).

17 Series for /π 7 0. Jesús Guillera A discrete function An, k) is hypergeometric if An +, k) An, k) and An, k + ) An, k) are both rational functions. A pair of functions F n, k) and Gn, k) is said to be a WZ pair after H. S. Wilf and D. Zeilberger ) if F and G are hypergeometric and F n +, k) F n, k) = Gn, k + ) Gn, k). In this case, H. Wilf and D. Zeilberger [53] showed that there exists a rational function Cn, k) such that Gn, k) = Cn, k)f n, k). The function Cn, k) is called a certificate of F, G). Defining Hn, k) = F n + k, n + ) + Gn, n + k), Wilf and Zeilberger showed that Hn, 0) = n=0 Gn, 0). Ekhad Zeilberger s computer) and Zeilberger [44] were the first to use this method to derive a one page proof of the representation 2 π = ) 3 ) )!). 0.) 3 The identity 0.) was first proved by G. Bauer in 859 [0]. Ramanuan recorded 0.) as Example 4 in Section 7 of Chapter 0 in his second notebook [57], [, pp ]. Further references can be found in [7]. Motivated by this work, J. Guillera [46] found many new WZ-pairs F, G) and derived new series not only for /π but for /π 2 as well. One of his most elegant formulas is 28 π 2 = 2 ) n=0 ) ) 2 0. Subsequently, Guillera empirically discovered many series of the type A π = B 2 + D + E c. H Most of the series he discovered cannot be proved by the WZ-method; it appears that the WZ-method is only applicable to those series for /π when H is a power of 2. An example of Guillera s series which remains to be proved is [47] 28 5 π 2 = ) 2 5 ) 2 3) 3) 6) 6) ).!) For further Ramanuan-like series for /π 2, see Zudilin s papers [65], [66].

18 8 Baruah, Berndt, and Chan. Recent Developments We have emphasized in this paper that Clausen s formula 6.) is an essential ingredient in most proofs of Ramanuan-type series representations for /π. However, there are other kinds of series for /π that do not depend upon Clausen s formula. One such series discovered by Takeshi Sato [60] is given by π ) µ = µ=0 ν=0 µ ν ) 2 µ + ν ν ) ) ) 2µ µ..) 20 2 In unpublished work, Sato [6] derived a more complicated series for /π that yields approximately 97 digits of π per term. A companion to.), which was derived by a new method devised by the third author, S. H. Chan, and Z. G. Liu [30], is given by where a µ := 8 3π = µ ν=0 µ=0 2µ 2ν µ ν a µ 5µ + 64 µ,.2) ) 2ν ν ) ) 2 µ..3) ν We cite three further new series arising from this new method. The first is another companion of.2), which arises from recent work of Chan and H. Verrill [34] after the work of Almkvist and Zudilin [2]), and is given by 9 2 3π = µ 3 ) µ µ + ν ) µ ν 3 µ 3ν 3ν ν µ=0 ν=0 ) ) µ 3ν)! 4µ + ). ν!) 3 8 The second is from a paper by Chan and K. P. Loo [33] and takes the form ) = C µ µ + 2 ) ) µ 2, 9π 3 4 where C µ = ν=0 µ=0 { µ ν ) 3 µ ν ν ) } 3 µ ν. i The third was derived by Y. Yang [63] and takes the shape 8 µ ) 4 µ π 5 = 4µ +. ν 36 ν µ=0 ν=0 Arising out of his work with Mahler measures and new transformation formulas for 5F 4 series, M. D. Rogers [59, Corollary 3.2] has also discovered series for /π in the i=0

19 Series for /π 9 spirit of the formulas above. For example, if a µ is defined by.3), then 2 π = ) µ 3µ + a µ 32. µ µ=0 This series was also independently discovered by Chan and Verrill [34]. According to W. Zudilin [67], G. Gourevich, using an integers relation algorithm, empirically discovered a hypergeometric-like series for /π 3, namely, 32 π = 3 µ=0 2 ) 7 µ µ!) 7 2 6µ 68µ3 + 76µ 2 + 4µ + ). This series and the search for further series representations for /π m, m 2, are described in a paper by D. H. Bailey and J. M. Borwein [4]. We mentioned briefly in Section 5 that 5.) could be used to determine any particular binary digit of π. It is likely that Ramanuan s series for /π can be used for the same purpose. To that end, let N+t 2 3) 3) 2) P N,t = ) ). 3!) Then it appears that the 6N 5)-th to 6N-th decimal places of /π can be obtained from the first six decimal places of 0 6N ) P N,[N/6]. For example, when N = 2, we can obtain the 7-th to 2-th decimal places of /π from the first six decimal places of Conclusion One test of good mathematics is that it should generate more good mathematics. Readers have undoubtedly concluded that Ramanuan s original series for /π have sown the seeds for an abundant crop of good mathematics. 3. Acknowledgments The authors are deeply grateful to R. William Gosper for carefully reading earlier versions of this paper, uncovering several errors, and offering excellent suggestions. We are pleased to thank Si Min Chan and Si Ya Chan for watching High School Musical, thereby making their father aware of Walt Disney Productions interest in Ramanuan s formulas for /π. References [] G. Almkvist, C. Krattenthaler, and J. Petersson, Some new formulas for π, Experiment. Math ), [2] G. Almkvist and W. Zudilin, Differential equations, mirror maps and zeta values, in Mirror Symmetry V, N. Yui, S. T. Yau, and J. D. Lewis, eds., AMS/IP Studies in Advanced Mathematics 38, American Mathematical Society, Providence, RI, 2006, pp [3] G. E. Andrews and B. C. Berndt, Ramanuan s Lost Notebook, Part II, Springer, New York, to appear.

20 20 Baruah, Berndt, and Chan [4] D. H. Bailey and J. M. Borwein, Computer-assisted discovery and proof, in Tapas in Experimental Mathematics, Contemp. Math., American Mathematical Society, Providence, RI, to appear. [5] W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, London, 935. [6] A. Baker, Transcendental Number Theory, Cambridge University Press, Cambridge, 975. [7] N. D. Baruah and B. C. Berndt, Eisenstein series and Ramanuan-type series for /π, submitted for publication. [8] N. D. Baruah and B. C. Berndt, Ramanuan s series for /π arising from his cubic and quartic theories of elliptic functions, J. Math. Anal. Applics ), [9] N. D. Baruah and B. C. Berndt, Ramanuan s Eisenstein series and new hypergeometric-like series for /π 2, J. Approx. Thy., to appear. [0] G. Bauer, Von den Coefficienten der Reihen von Kugelfunctionen einer Variabeln, J. Reine Angew. Math ), 0 2. [] B. C. Berndt, Ramanuan s Notebooks, Part II, Springer-Verlag, New York, 989. [2] B. C. Berndt, Ramanuan s Notebooks, Part III, Springer-Verlag, New York, 99. [3] B. C. Berndt, Ramanuan s Notebooks, Part IV, Springer-Verlag, New York, 994. [4] B. C. Berndt, Ramanuan s Notebooks, Part V, Springer-Verlag, New York, 998. [5] B. C. Berndt, S. Bhargava and F. G. Garvan, Ramanuan s theories of elliptic functions to alternative bases, Trans. Amer. Math. Soc ), [6] B. C. Berndt and H. H. Chan, Notes on Ramanuan s singular moduli, in Number Theory, Fifth Conference of the Canadian Number Theory Association, R. Gupta and K. S. Williams, eds., American Mathematical Society, Providence, RI, 999, pp [7] B. C. Berndt and H. H. Chan, Ramanuan and the modular invariant, Canad. Math. Bull ), [8] B. C. Berndt and H. H. Chan, Eisenstein series and approximations to π, Illinois J. Math ), [9] B. C. Berndt, H. H. Chan, and W. C. Liaw, On Ramanuan s quartic theory of elliptic functions, J. Number Thy ), [20] B. C. Berndt, H. H. Chan, and L. C. Zhang, Ramanuan s singular moduli, Ramanuan J. 997), [2] B. C. Berndt and R. A. Rankin, Ramanuan: Essays and Surveys, American Mathematical Society, Providence, RI, 200; London Mathematical Society, London, 200. [22] J. M. Borwein and P. B. Borwein, Pi and the AGM; A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 987. [23] J. M. Borwein and P. B. Borwein, Ramanuan s rational and algebraic series for /π, J. Indian Math. Soc ), [24] J. M. Borwein and P. B. Borwein, More Ramanuan-type series for /π, in Ramanuan Revisited, G. E. Andrews, R. A. Askey, B. C. Berndt, K. G. Ramanathan, and R. A. Rankin, eds., Academic Press, Boston, 988, pp [25] J. M. Borwein and P. B. Borwein, Ramanuan and Pi, Scientific American 256, February, 988, 2 7; reprinted in [2], pp [26] J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi s identity and the AGM, Trans. Amer. Math. Soc ), [27] J. M. Borwein and P. B. Borwein, Some observations on computer aided analysis, Notices Amer. Math. Soc ), [28] J. M. Borwein and P. B. Borwein, Class number three Ramanuan type series for /π, J. Comput. Appl. Math ), [29] J. M. Borwein, P. B. Borwein, and D. H. Bailey, Ramanuan, modular equations, and approximations to pi or how to compute one billion digits of pi, Amer. Math. Monthly ), [30] H. H. Chan, S. H. Chan, and Z. Liu, Domb s numbers and Ramanuan-Sato type series for /π, Adv. in Math ), [3] H. H. Chan and W. C. Liaw, Cubic modular equations and new Ramanuan-type series for /π, Pacific J. Math ),

21 Series for /π 2 [32] H. H. Chan, W. C. Liaw, and V. Tan, Ramanuan s class invariant λ n and a new class of series for /π, J. London Math. Soc. 2) ), [33] H. H. Chan and K. P. Loo, Ramanuan s cubic continued fraction revisited, Acta Arith ), [34] H. H. Chan and H. Verrill, The Apéry numbers, the Almkvist-Zudilin numbers and new series for /π, in preparation. [35] S. Chowla, Series for /K and /K 2, J. London Math. Soc ), 9 2. [36] S. Chowla, On the sum of a certain infinite series, Tôhoku Math. J ), [37] S. Chowla, The Collected Papers of Sarvadaman Chowla, Vol., Les Publications Centre de Recherches Mathématiques, Montreal, 999. [38] D. V. Chudnovsky and G. V. Chudnovsky, Approximation and complex multiplication according to Ramanuan, in Ramanuan Revisited, G. E. Andrews, R. A. Askey, B. C. Berndt, K. G. Ramanathan, and R. A. Rankin, eds., Academic Press, Boston, 988, pp [39] D. V. Chudnovsky and G. V. Chudnovsky, The computation of classical constants, Proc. Nat. Acad. Sci. U.S.A ), [40] D. V. Chudnovsky and G. V. Chudnovsky, Computational problems in arithmetic of linear differential equations. Some diophantine applications, in Number Theory, New York , Lecture Notes in Mathematics No. 383, D. V. Chudnovsky, G. V. Chudnovsky, H. Cohn, and M. B. Nathanson, eds., Springer-Verlag, Berlin, 989, pp [4] D. V. Chudnovsky and G. V. Chudnovsky, Classical constants and functions: Computations and continued fraction expansions, in Number Theory, New York Seminar , D. V. Chudnovsky, G. V. Chudnovsky, H. Cohn, and M. B. Nathanson, eds., Springer-Verlag, Berlin, 99, pp [42] D. V. Chudnovsky and G. V. Chudnovsky, Hypergeometric and modular function identities, and new rational approximations to and continued fraction expansions of classical constants and functions, in A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemporary Mathematics, No. 43, M. Knopp and M. Sheingorn, eds., American Mathematical Society, Providence, RI, 993, pp [43] D. V. Chudnovsky and G. V. Chudnovsky, Generalized hypergeometric functions classification of identities and explicit rational approximations, in Algebraic Methods and q-special Functions, CRM Proceedings and Lecture Notes, Vol. 22, American Mathematical Society, Providence, RI, 993, pp [44] S. Ekhad and D. Zeilberger, A WZ proof of Ramanuan s formula for π, in Geometry, Analysis and Mechanics, World Scientific, River Edge, NJ, 994, pp [45] R. W. Gosper, Jr., File of correspondence on the author s computation of π. [46] J. Guillera, Some binomial series obtained by the WZ-method, Adv. Appl. Math ), [47] J. Guillera, About a new kind of Ramanuan-type series, Experiment. Math ), [48] J. Guillera, Generators of some Ramanuan formulas, Ramanuan J. 2006), [49] J. Guillera, A new method to obtain series for /π and /π 2, Experiment. Math ), [50] J. Guillera, A class of conectured series representations for /π, Experiment. Math ), [5] J. Guillera, Hypergeometric identities for 0 extended Ramanuan type series, Ramanuan J., to appear. [52] M. Hata, Rational approximations to π and some other numbers, Acta Arith ), [53] M. Petkovsek, H. S. Wilf, and D. Zeilberger, A = B, AK Peters, Wellesey, MA, 996. [54] R. Preston, Profiles: the mountains of pi, The New Yorker, March 2, 992, pp [55] S. Ramanuan, Modular equations and approximations to π, Quart. J. Math. Oxford) 45 94),

22 22 Baruah, Berndt, and Chan [56] S. Ramanuan, Collected Papers, Cambridge University Press, Cambridge, 927; reprinted by Chelsea, New York, 962; reprinted by the American Mathematical Society, Providence, RI, [57] S. Ramanuan, Notebooks 2 volumes), Tata Institute of Fundamental Research, Bombay, 957. [58] S. Ramanuan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 988. [59] M. D. Rogers, New 5 F 4 hypergeometric transformations, three-variable Mahler measures, and formulas for /π, Ramanuan J., to appear. [60] T. Sato, Apéry numbers and Ramanuan s series for /π, Abstract for a lecture presented at the annual meeting of the Mathematical Society of Japan, 28 3 March [6] T. Sato, Personal communication to the third author. [62] H. Weber, Lehrbuch der Algebra, dritter Band, Chelsea, New York, 96. [63] Y. Yang, Personal communication. [64] W. Zudilin, Ramanuan-type formulae and irrationality measures of certain multiples of π, Mat. Sb ), [65] W. Zudilin, Quadratic transformations and Guillera s formulae for /π 2, Math. Zametki ), Russian); Math. Notes ), [66] W. Zudilin, More Ramanuan-type formulae for /π 2, Russian Math. Surveys ). [67] W. Zudilin, Ramanuan-type formulae for /π: A second wind?, in Modular Forms and String Duality Banff, June, 2006), N. Yui, ed., London Math. Soc. Lecture Notes, to appear. Department of Mathematical Sciences, Tezpur University, Napaam , Sonitpur, Assam, India address: nayan@tezu.ernet.in Department of Mathematics, University of Illinois, 409 West Green St., Urbana, IL 680, USA address: berndt@math.uiuc.edu Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 9260, Republic of Singapore address: matchh@nus.edu.sg