Some theorems on the explicit evaluations of singular moduli 1

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1 General Mathematics Vol 17 No 1 009) Some theorems on the explicit evaluations of singular moduli 1 K Sushan Bairy Abstract At scattered places in his notebooks Ramanujan recorded some theorems for calculating singular moduli and also recorded several values of singular moduli In this paper we establish several general theorems for the explicit evaluations of singular moduli We also obtain some values of class invariants and singular moduli 000 Mathematics Subject Classification: Primary D10 11F7 Key words and phrases: Singular Moduli Modular equation Class invariant 1 Introduction In Chapter 16 of his second notebook [] [10] Ramanujan has defined his theta-function as fab) : n a nn+1)/ b nn1)/ ab < 1 1 Received May 008 Accepted for publication in revised form) 10 December

2 7 K Sushan Bairy Following Ramanujan we define where where a;q) : χq) : q;q ) 1 aq n ) q < 1 n0 The ordinary hypergeometric series F 1 ab;c;x) is defined by F 1 ab;c;x) n0 a) n b) n c) n n! xn a) 0 1 a) n aa + 1)a + )a + n 1) for n 1 x < 1 The complete elliptic integral of the first kind K : Kα) associated with the modulus α 0 < α < 1 is defined by Kα) : π 0 dφ 1 α sin φ π 1 F 1 1 ) ; 1;α where the latter representation is achieved by expanding the integrand in a binomial series and integrating termwise Singular modulus α : α n is that unique positive number between 0 and 1 satisfying 1 F 1 11) 1; 1; 1 α ) n n 1 F 1 1; 1;α ) ; n Q + n At scattered places in his first notebook Ramanujan recorded several values of singular moduli α n : αe πn ) in terms of units On page 8 of his first notebook he gave some theorems for calculating α n when n is even J M Borwein and P B Borwein [6] had calculated some of Ramanujan s values for α n Watson [1] used the formula found in Ramanujan s first notebook [10 vol1 p 0] to prove the value of k 10 where α n k n which Ramanujan wrote in his second letter to Hardy [11 p xxix]

3 Some theorems on the explicit 7 and Let 1 Zr) : Zr;x) F 1 r r 1 ) ; 1;x r q r : q r x) : exp π csc where r 6 and 0 < x < 1 π r ) F 1 1 r r1 r ) 1; 1 x) F 1 1 r r1 r 1;x) Let n denote a fixed natural number and assume that 1) n F 1 1 r1 ; 1; 1 α) r r 1 F 1 r1; 1;α) F 1 1 r1 ; 1; 1 β) r r 1 r r F 1 r1; 1;β) r r where r or 6 Then a modular equation of degree n in theory of elliptic functions of signature r is a relation between moduli α and β induced Zr : α) by 1) We often say that β is of degree n over α and mr) : Zr : β) is called multiplier We also use the notations Z 1 : Z 1 r) Zr : α) and Z n : Z n r) Zr : β) to indicate that β has degree n over α When the context is clear we omit the argument r in q r Zr) and mr) Ramanujan class invariants are defined as 1) G n : 1/ q 1/ χq) {α1 α)} 1/ and 1) g n : 1/ q 1/ χq) {α1 α) } 1/ On pages 9 99 Ramanujan recorded table of values for 77 class invariants or monic irreducible polynomials In [1] [1] Watson proved of Ramanujan s class invariants from Ramanujan s paper [10] Watson also wrote further four papers [15] [16] [17] [18] on the calculation of class invariants In [] B C Berndt H H Chan and L -C Zhang has used class field theory Galois theory and Kronecker s limit formula to justify Watson s assumptions and calculated some values of G n In [1] N D Baruah has established the value of G 17 In [9] Mahadeva Naika and K Sushan Bairy

4 7 K Sushan Bairy have established several new explicit evaluations of class invariants and singular moduli In [8] Mahadeva Naika has established several new explicit values for G n In this paper we establish several general formulas for explicit evaluations of singular moduli using Ramanujan s modular equations We also obtain some values of class invariant and singular moduli Preliminary Results In this section we collect the identities which are useful in proving our main results Lemma 1 If β is of degree over α then 1) αβ) 1/ + {1 α)1 β)} 1/ 1 For a proof of Lemma 1 see Entry 5 of Chapter 19 in [ pp 0 1] Lemma If β is of degree 5 over α then ) αβ) 1/ + {1 α)1 β)} 1/ + {16αβ1 α)1 β)} 1/6 1 For a proof of Lemma see Entry 1 of Chapter 19 in [ pp 80 8] Lemma If β is of degree 7 over α then ) αβ) 1/8 + {1 α)1 β)} 1/8 1 For a proof of Lemma see Entry 19 of Chapter 19 in [ pp 1 15] Lemma If β is of degree 11 over α then ) αβ) 1/ + {1 α)1 β)} 1/ + {αβ1 α)1 β)} 1/1 1 For a proof of Lemma see Entry 7 of Chapter 0 in [ p 6]

5 Some theorems on the explicit 75 Lemma 5 If β is of degree over α then 5) αβ) 1/8 + {1 α)1 β)} 1/8 + / {αβ1 α)1 β)} 1/ 1 For a proof of Lemma 5 see Entry 15 of Chapter 0 in [ p 11] Lemma 6 We have 6) g 1 n 1 αn α n Using 1) we obtain 6) Lemma 7 If a qb d a perfect square then 7) a + b a + d a d q + sgn b) The above lemma is due to Bruce Reznick For a proof of Lemma 7 via Chebyshev polynomials see [5 p 150] Lemma 8 We have 8) [ gng 9n + 1 ] g6 9n gng 9n gn 6 ] 9) [ G ng 9n 1 [ 10) G ng 5n G ng 9n ] 1 G6 9n G 6 n g6 n g9n 6 + G6 n G 6 9n + G n G 5n G 5n G ng 5n G n Proofs can be found in [ pp 1 8] [7 Th1) )] Lemma 9 We have G ng g 9n ng9n + ) gng 9n + gn g9n 11) gng 9n G ng 9n + ) G ng 9n G n G 9n 1) ) 1) gng 6n gng 9n gng 9n + gng 9n + gn g9n ) 1) G ng 9n G ng 9n + G ng 9n G n G 9n

6 76 K Sushan Bairy Proofs can be found in [7] Main Theorems In this section we obtain several general formulas connecting singular moduli and class invariants Theorem 1 We have G G α 9n G 1 1 n + 1 n n + 1 1) u 1 8 u ) gn 1 + ) [ + v 6 gn + 1 ] 8 v 6 where u G n G 9n ) 1 and v g n g 9n Proof of 1) Using 1) in 1) we find that [ ] 1 1 u 6 ) αβ Using 1) and 7) in the above equation ) we obtain the required result Proof of ) Using 1) in 1) we find that ) αβ [ v + v 6 + ] 8 Using 1) and 7) in the above equation ) we obtain the required result

7 Some theorems on the explicit 77 Corollary 1 We have α 7 1/ + ) ) 8 5) 1 1 Proof Using n in 1) with 6) 7) G 1/1 ) 1/ G 7 1/1 1 we obtain the required result Theorem We have G α 5n G 1 1 n + 1 8) n G 1 n [ 1 u + u 6 ] 1 u u 6 where u G n G 5n ) 1 Proof Using 1) in ) we deduce that [ 1 u ) ] 1 u 9) αβ ) u 1 Using 1) and 7) in 9) we deduce the required result Corollary We have 10) α ) 1/ ) 15 Proof From [ p191] we have 11) G 5 + ) ) 1/ 1/ + 5 5

8 78 K Sushan Bairy Using 11) with n 9 5 in 10) we deduce that 1) G9 5 + ) ) 1/ 1/ 5 5 Using 11) and 1) with n 9 in 8) we obtain the required result 5 Remark: A different proof of α 5 can be found in [ p 90] Theorem We have G G α 9n G 1 1 n + 1 n n + 1 1) 1 8 [ 1 1 ] 8 u gn 1 + ) gn v v 1) where u G n G 9n ) 1 and v g n g 9n Proof of 1) Using 1) in ) we find that [ 1 1 ] 8 u 15) αβ Using 1) and 7) in equation 15) we obtain the required result 1) Proof of 1) Using 1) and 7) in ) we find that 16) αβ + v v Using 1) and 7) in the above equation 16) we obtain the required result 1) 16 16

9 Some theorems on the explicit 79 Corollary We have 17) α ) 8 7 Proof From [ p 191] we have 7 1/ + + ) 7 18) G 9 Using n 1 with 18) in 1) we obtain the required result Corollary We have 19) α [ ] 16 Proof From [ p 00] we have ) g 1 Using n 7 1) g 7 in 8) we deduce that Using 0) and 1) with n in 1) we obtain the required result 7

10 80 K Sushan Bairy Theorem We have G G α 11n G 1 1 n + 1 n n + 1 ) u + u 1 u 8 u ) gn 1 + [ ) vv + ) + ] 8 v gn + 1 v + ) + where u G n G 11n ) 1 and v g n g 11n Proof of ) Using 1) and 7) in ) we deduce that ) αβ 1 u + u 1 u 8 u Using 1) and 7) in ) we deduce the required result ) Proof of ) Using 1) in ) we deduce that [ vv + ) + ] 8 v 5) αβ v + ) + Using 1) in the above equation 5) we find the required result ) Theorem 5 We have G α 59n G 1 1 n + 1 6) n ) G 1 n u) + 1 u) 8 u gn 1 + ) gn + 1 vv + vv ) + + ) + 16

11 Some theorems on the explicit 81 where u G n G 59n ) 1 and v g n g 59n Proof of 6) Using 1) in 5) we deduce that 8) αβ 1 u) + 1 u) u Using 1) and 7) in 8) we obtain the required result 6) Proof of 7) Using 1) in 5) we deduce that 9) αβ vv + vv ) + + ) + Using 1) in the above equation 9) we find the required result 7) 8 8 Explicit Evaluations of g n In this section we obtain some explicit evaluations of g n Theorem 1 We have ) 1/ g ) ) g8 ) 1/ Proof In [7 Th5] we have ) g8g + 1 Using ) in 8) we deduce that ) g 6 g From ) and ) we obtain the required results 1/1 1/1

12 8 K Sushan Bairy Theorem We have 5) g ) 1/ ) 1/ /1 6) g 1 + ) 1/ ) 1/ /1 ) 1/ ) 1/ ) 1/ g 0 1/ ) /1

13 Some theorems on the explicit 8 ) 1/ ) 1/ ) 1/ g80 1/ ) /1 Proof of 5) Using ) in 1) we obtain the required result Since the proofs of 6) 8) are similar to the proof of 5) we omit the details 5 Explicit Evaluations of G n In this section we obtain some explicit evaluations of G n Theorem 51 We have ) 1/ G ) /1 1/1

14 8 K Sushan Bairy G8 5) ) 1/ /1 1/1 Theorem 51) is obtained by using Theorem 1) and the formula g n 1 G n g n 6 Explicit Evaluations of α n In this section we obtain some explicit evaluations of α n Theorem 61 We have 61) α ) ) 6) α )

15 Some theorems on the explicit 85 6) ) 1 + α ) 1 6) ) 1 α ) 1 1 Proofs are similar to the proof of corollary 1 so we omit the details Acknowledgement I wish to thank Dr M S Mahadeva Naika for his valuable suggestions and encouragement during the preparation of this paper I also wish to thank the referee for their valuable suggestions References [1] N D Baruah On some class invariants of Ramanujan J Indian Math Soc )

16 86 K Sushan Bairy [] B C Berndt Ramanujan s Notebooks Part III Springer-Verlag New York 1991 [] B C Berndt Ramanujan s Notebooks Part V Springer-Verlag New York 1998 [] B C Berndt H H Chan and L-C Zhang Ramanujan s class invariants Kronecker s limit formula and modular equations Trans Amer Math Soc 96) ) [5] B C Berndt H H Chan and L-C Zhang Radicals and units in Ramanujan s work Acta Arith 87) ) [6] J M Borwein and P B Borwein Pi and AGM Wiley New York 1987 [7] M S Mahadeva Naika P-Q eta-function identities and computation of Ramanujan-Weber class invariants Journal of the Indian Math Soc 701-) ) [8] M S Mahadeva Naika Some new explicit values for Ramanujan class invariants communicated) [9] M S Mahadeva Naika and K Sushan Bairy On some new explicit evaluations of class invariants Vietnam J Math 61) ) [10] S Ramanujan Notebooks volumes) Tata Institute of Fundamental Research Bombay 1957 [11] S Ramanujan Collected Papers Chelsea New York 196 [1] G N Watson Theorems stated by Ramanujan XII): A singular modulus J London Math Soc ) [1] G N Watson Some Singular modulii) Quart J Math )

17 Some theorems on the explicit 87 [1] G N Watson Some Singular moduliii) Quart J Math ) [15] G N Watson Singular moduli) Proc London Math Soc ) [16] G N Watson Singular moduli) Acta Arith ) [17] G N Watson Singular moduli5) Proc London Math Soc ) [18] G N Watson Singular moduli6) Proc London Math Soc ) Department of Mathematics Central College Campus Bangalore University Bangalore INDIA ksbairy@rediffmailcom