## Ramanujan's Association With Radicals In India (1997)

### BibTeX

@MISC{Berndt97ramanujan'sassociation,

author = {Bruce C. Berndt and Heng Huat Chan and Liang-Cheng Zhang},

title = {Ramanujan's Association With Radicals In India},

year = {1997}

}

### OpenURL

### Abstract

this paper and in calculations of further class invariants. The value of G 1353 was communicated by Ramanujan [16, p. xxix, eq. (23)], [8, p. 62] in his second letter, dated 27 February 1913, to G. H. Hardy and was first established (unrigorously) by G. N. Watson [18]. In a letter of 1 October 1930 to B. M. Wilson [8, pp. 237, 238], Watson confided, " : : : but 23 which deals with the singular modulus associated with 1353 is included; I was pleased at getting this out, because the bulk of the singular moduli in the Notebooks can be obtained in the same way : : : You will be interested to hear how Ramanujan got no. 23, particularly when you look at the length of the answer. I am absolutely convinced that he guessed it." (Calculating a singular modulus, which we do not define here, is equivalent to calculating a class invariant.) The reader is undoubtedly astonished to learn that Ramanujan first "guessed" his formula for G 1353 : We do not agree with Watson! We think that Watson's proof, which is not rigorous, could not have been given without his knowing the formula in advance. The first rigorous proof was given recently by Chan [9]. On pages 294--299 in his second notebook [15], Ramanujan gave a table of values for 77 class invariants, three of which are not found in the first notebook. Since the second notebook is an enlarged revision of the first, it is unclear why Ramanujan failed to record 33 class invariants that he offered in the first notebook. Four further results are found in scattered places in the second notebook. After arriving in Cambridge, Ramanujan learned of Weber's work [21], and so when he wrote his paper [14], [16, pp. 23--39], the table of 46 class invariants that he included did not contain any that are found in Weber's book [21]. Except for G 325 a...

### Citations

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Ramanujan’s Notebooks Part V
- Berndt
- 1998
(Show Context)
Citation Context ...mials satisfied by them. The excellent text of D. A. Cox [11] provides an accessible account of Weber’s work on invariants. Before proceeding further, we give some examples that Ramanujan calculated: =-=(2)-=- G5 = G17 = � 1 + √ 5 2 � 1/4 � 5 + √ � √17 17 − 3 + , 8 8 � 5 + (3) G69 = √ �1/12 � 23 3 √ 2 √ 3 + √ � ⎛� 1/8 23 ⎝ 6 + 3 2 √ 3 4 and G1353 = � 3 + √ �1/4 � 11 √ 2 ⎛� × ⎝ 17 + 3 √ 33 8 5 + 3 √ �1/4 � ... |

171 |
The lost notebook of other unpublished papers
- Ramanujan
- 1988
(Show Context)
Citation Context ...1/24 f(−q) q f(−q 2 ) , when q = exp(−π √ n). Ramanujan likely used his values of class invariants to calculate explicitly certain products of eta–functions in both his first and lost notebooks [15], =-=[17]-=-. For example, he probably used the values of G225 and G 25/9 to prove that (9) e 6π/5 f(−e−6π/5 ) f(−e −30π ) = a + b a − b , where a = (60) 1/4 and b = 2 − √ 3 + √ 5. Ramanujan also used class invar... |

113 |
Primes of the form x 2 + ny 2
- Cox
- 1989
(Show Context)
Citation Context ...rhaps for these reasons, Weber called Gn and gn class invariants, and computed a total of 105 class invariants or the monic, irreducible polynomials satisfied by them. The excellent text of D. A. Cox =-=[11]-=- provides an accessible account of Weber’s work on invariants. Before proceeding further, we give some examples that Ramanujan calculated: (2) G5 = G17 = � 1 + √ 5 2 � 1/4 � 5 + √ � √17 17 − 3 + , 8 8... |

85 | Notebooks (2 Volumes), Tata Institute of Fundamental Research - Ramanujan - 1957 |

68 |
Collected Papers
- Ramanujan
- 1962
(Show Context)
Citation Context ...er another example [4], Ramanujan undoubtedly used (9) to show that where R(e −6π ) = � c 2 + 1 − c, 2c := 1 + a + b√ 5. a − b Ramanujan calculated several further values of R(q) in his lost notebook =-=[16]-=-, and many of these can be found in the authors’ paper [6]. In the remainder of the paper, we briefly describe some attempts and methods used to establish Ramanujan’s class invariants. In two papers [... |

44 |
Modular equations and approximations to π, Quart
- Ramanujan
- 1914
(Show Context)
Citation Context ...interest in radical equalities merely a consequence of their popularity in his time, or were there other reasons? The answer can be found in his notebooks [15] and in one of his most important papers =-=[14]-=-, [16, pp. 23–39]. Scattered among the pages in Ramanujan’s first notebook are the values of 107 class invariants, or polynomials satisfied by them. As we shall see, these invariants frequently take t... |

37 | Primes of the Form x + ny - Cox - 1989 |

24 |
Ramanujan
- Berndt, Rankin
- 1995
(Show Context)
Citation Context ...= exp(2πiz), where |q| < 1. The exponents k(3k − 1)/2 are called pentagonal numbers, and the second equality in (7) constitutes Euler’s pentagonal number theorem. From (1) and (7), we easily see that =-=(8)-=- Gn = 2 −1/4 −1/24 f(q) q f(−q 2 ) and gn = 2 −1/4 −1/24 f(−q) q f(−q 2 ) , when q = exp(−π √ n). Ramanujan likely used his values of class invariants to calculate explicitly certain products of eta–f... |

16 |
Explicit evaluations of the RogersRamanujan continued fraction
- Berndt, Chan, et al.
- 1996
(Show Context)
Citation Context ...how that where R(e −6π ) = � c 2 + 1 − c, 2c := 1 + a + b√ 5. a − b Ramanujan calculated several further values of R(q) in his lost notebook [16], and many of these can be found in the authors’ paper =-=[6]-=-. In the remainder of the paper, we briefly describe some attempts and methods used to establish Ramanujan’s class invariants. In two papers [19], [20], Watson proved the 24 class invariants from Rama... |

15 |
Some applications of Kronecker’s limit formula
- Ramanathan
- 1987
(Show Context)
Citation Context ...s to representations for certain products of Dedekind eta–functions in terms of fundamental units. By (8), these representations allow us to calculate Gn. Our methods extend those of K. G. Ramanathan =-=[13]-=- who calculated some of Ramanujan’s class invariants but required that Q( √ −n) contains only one class per genus. Zhang [22], [23] has further extended the method to give rigorous proofs of the invar... |

14 | algebras - Hall - 1990 |

14 | Modular equations and approximations to r, Quart - Ramanujan - 1914 |

13 |
Ramanujan’s class invariants, Kronecker’s limit formula, and modular equations
- Berndt, Chan, et al.
- 1997
(Show Context)
Citation Context ...+ 3 √ 33 8 � , 11 + √ �1/4 � 123 √ 2 ⎞ ⎠ 1/2 569 + 99 √ 33 8 ⎞ ⎠ 1/2 . + � 2 + 3 √ 3 4 ⎞ ⎠ 1/2 6817 + 321 √ �1/12 451 √ 2 The value of G69 was only recently verified for the first time by the authors =-=[7]-=-. In our calculation of G69, we used the equality � 188 + 108 √ � 3 + (188 + 108 √ 3) 2 � �1/6 6 + 3 − 1 = √ � 3 2 + 3 + 4 √ 3 , 4 ,sRAMANUJAN AND RADICALS 3 which is the special case, a = (4 + 3 √ 3)... |

12 | Some values for the Rogers-Ramanujan continued fraction
- Berndt, Chan
- 1995
(Show Context)
Citation Context ...er theory, perhaps you have shown that √ 5 − 1 R(1) = . 2 Using the value of G5, given in (2), we can show that R 5 (e −2π/√ � �� � 5 ) = � 5 √ �2 5 + 11 + 1 − 2 5√5 + 11 . 2 To offer another example =-=[4]-=-, Ramanujan undoubtedly used (9) to show that where R(e −6π ) = � c 2 + 1 − c, 2c := 1 + a + b√ 5. a − b Ramanujan calculated several further values of R(q) in his lost notebook [16], and many of thes... |

12 |
Some singular moduli (i
- Watson
- 1932
(Show Context)
Citation Context ...], and many of these can be found in the authors’ paper [6]. In the remainder of the paper, we briefly describe some attempts and methods used to establish Ramanujan’s class invariants. In two papers =-=[19]-=-, [20], Watson proved the 24 class invariants from Ramanujan’s paper [14] that cannot be found in Ramanujan’s second notebook. In the first [19], Watson devised an “empirical process” to calculate 14 ... |

12 |
Lehrbuch der Algebra, dritter
- Weber
- 1961
(Show Context)
Citation Context ...ain even values of n. 1 th Typeset by AMS-TEXs2 BRUCE C. BERNDT, HENG HUAT CHAN, AND LIANG–CHENG ZHANG At the beginning of the twentieth century, these invariants were extensively studied by H. Weber =-=[21]-=-, who used the notations Gn =: 2−1/4f( √ −n) and gn =: 2−1/4f1( √ −n). Weber [21] proved that Gn and gn are algebraic. In fact, Gn, 2−1/12Gn, and 2−1/4Gn are units in some algebraic number field accor... |

11 |
Ramanujan’s class invariants and cubic continued fraction
- Berndt, Chan, et al.
- 1995
(Show Context)
Citation Context ...the familiar trigonometric functions which have just one linearly independent period. The complete elliptic integral of the first kind associated with the modulus k, 0 < k < 1, is defined by � π/2 dθ =-=(5)-=- K := K(k) := � . 1 − k2 2 sin θ 0 The complementary modulus k ′ is defined by k ′ = √ 1 − k2 ; set K ′ = K(k ′ ). If q = exp(−πK ′ /K), then one of the central theorems in the theory of elliptic func... |

10 | Ramanujan’s explicit values for the classical thetafunction, Mathematika 42
- Berndt, Chan
- 1995
(Show Context)
Citation Context ...manujan used class invariants to determine explicitly particular values of the theta function ϕ(q) defined by ϕ(q) := ∞� k=−∞ q k2 . For example, Ramanujan probably used his value of G49 to show that =-=[3]-=- ϕ (4) 2 (e−7π ) ϕ2 (e−π ) = � √ � √ 13 + 7 + 7 + 3 7 (28) 14 1/8 .s4 BRUCE C. BERNDT, HENG HUAT CHAN, AND LIANG–CHENG ZHANG The value ϕ(e −π ) = π1/4 Γ( 3 4 ) is well known [1, p. 103], and so (4) pr... |

8 |
Theorems stated by Ramanujan (XIV): A singular modulus
- Watson
- 1931
(Show Context)
Citation Context ... value of G1353 was communicated by Ramanujan [16, p. xxix, eq. (23)], [8, p. 62] in his second letter, dated 27 February 1913, to G. H. Hardy and was first established (unrigorously) by G. N. Watson =-=[18]-=-. In a letter of 1 October 1930 to B. M. Wilson [8, pp. 237, 238], Watson confided, “ . . . but 23 which deals with the singular modulus associated with 1353 is included; I was pleased at getting this... |

6 |
Ramanujan--Weber class invariant Gn and Watson's empirical process
- Chan
- 1998
(Show Context)
Citation Context ...o not agree with Watson! We think that Watson’s proof, which is not rigorous, could not have been given without his knowing the formula in advance. The first rigorous proof was given recently by Chan =-=[9]-=-. On pages 294–299 in his second notebook [15], Ramanujan gave a table of values for 77 class invariants, three of which are not found in the first notebook. Since the second notebook is an enlarged r... |

4 |
Kronecker's limit formula, class invariants and modular equations (II), Analytic Number Theory
- Zhang
- 1997
(Show Context)
Citation Context ...ons allow us to calculate Gn. Our methods extend those of K. G. Ramanathan [13] who calculated some of Ramanujan’s class invariants but required that Q( √ −n) contains only one class per genus. Zhang =-=[22]-=-, [23] has further extended the method to give rigorous proofs of the invariants of Ramanujan that Watson [19] had “empirically” calculated. Our second method takes Watson’s ideas and employs class fi... |