Pacific Journal of Mathematics

Transcription

1 Pacific Journal of Mathematics A CERTAIN QUOTIENT OF ETA-FUNCTIONS FOUND IN RAMANUJAN S LOST NOTEBOOK Bruce C Berndt Heng Huat Chan Soon-Yi Kang and Liang-Cheng Zhang Volume 0 No February 00

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3 PACIFIC JOURNAL OF MATHEMATICS Vol 0 No 00 A CERTAIN QUOTIENT OF ETA-FUNCTIONS FOUND IN RAMANUJAN S LOST NOTEBOOK Bruce C Berndt Heng Huat Chan Soon-Yi Kang and Liang-Cheng Zhang In his lost notebook Ramanujan defined a parameter λ n by a certain quotient of Dedekind eta-functions at the argument q = exp π n/ He then recorded a table of several values of λ n To prove these values and others we develop several methods which include modular equations the modular j-invariant Kronecker s limit formula Ramanujan s cubic theory of elliptic functions and an empirical process Introduction On the top of the page in his lost notebook Ramanujan defined the function λ n by λ n = eπ/ n/ { e π n/ e π n/ e 4π } 6 n/ and then devoted the remainder of the page to stating several elegant values of λ n for n mod 8 namely: λ = λ 9 = λ 7 = 4 7 λ 5 = 5 λ = 8 λ 4 = 5 4 λ 49 = 55 λ 57 = λ 65 = λ 8 = λ 89 = λ 7 = λ 97 = λ = λ 69 = λ 9 = λ 7 = λ 4 = 6 67

4 68 BERNDT CHAN KANG AND ZHANG λ 65 = λ 89 = λ 6 = Note that for several values of n Ramanujan did not record the corresponding value of λ n The purpose of this paper is to establish all the values of λ n in including the ones that are not explicitly stated by Ramanujan by using the modular j-invariant modular equations Kronecker s limit formula and an empirical approach Applications of values for λ n will appear in papers by Chan W-C Liaw and V Tan [6] Chan A Gee and Tan [4] and by Berndt and Chan [6] The function λ n had been briefly introduced earlier in his third notebook [5 p 9] where Ramanujan offered a formula for λ n in terms of Klein s j-invariant first proved by Berndt and Chan [5] [ p 8 Entry ] by using Ramanujan s cubic theory of elliptic functions to alternative bases As KG Ramanathan [] pointed out the formula in the third notebook is for evaluating λ n/ especially for n = Observe that and 6 are precisely the discriminants congruent to 5 modulo 8 of imaginary quadratic fields of class number one Ramanathan inadvertently inverted the roles of n and n/ in his corresponding remark In Section we discuss some of these results in Ramanujan s third notebook and show how they can be used to calculate the values of λ n when n In this and the next two paragraphs we offer some necessary definitions Let ητ denote the Dedekind eta-function defined by ητ := e πiτ/4 e πinτ =: q /4 f q n= where q = e πiτ and Im τ > 0 Then can be written in the alternative form 6 i n/ 4 λ n = f 6 q qf 6 q = η η k=0 i n where q = e πn/ Since much of this paper is devoted to the evaluation of λ n by using modular equations we now give a definition of a modular equation Let a k = aa a k and define the ordinary hypergeometric function F a b; c; z by a k b k z k 5 F a b; c; z := z < c k k!

5 QUOTIENT OF ETA-FUNCTIONS 69 Suppose that 6 F ; ; β F ; ; β = n F ; ; α F ; ; α for some positive integer n A relation between the moduli α and β induced by 6 is called a modular equation of degree n and β is said to have degree n over α In Section using a modular equation of degree we derive a formula for λ n in terms of the Ramanujan-Weber class invariant which is defined by 7 G n := /4 q /4 q; q where q = exp π n and a; q = aq n n=0 In Sections 4 and 5 we establish all 8 values of λ p in the table Our proofs in Section 4 employ certain modular equations of degrees 5 7 and The first three were claimed by Ramanujan [5] and the last one was newly discovered by Berndt S Bhargava and F G Garvan [4] as a modular equation in the theory of signature in [4] In the theory of signature we say that the modulus β has degree n over the modulus α when 8 F ; ; β F ; ; β = n F ; ; α F ; ; α A modular equation of degree n in the theory of signature is a relation between α and β which is induced by 8 In Section 5 we employ recent discoveries of Chan and W C Liaw [5] [0] on Russell-type modular equations of degrees 7 and 9 in the theory of signature In these two sections we also determine the values of λ 5 λ 7 λ and λ 7 In two papers [] [] using Kronecker s limit formula Ramanathan determined several values of λ n In [] in order to determine two specific values of the Rogers Ramanujan continued fraction he evaluated λ 5 by applying Kronecker s limit formula to L-functions of orders of Q with conductor 5 This method was also used to determine λ 49 In the other paper [] Ramanathan found a representation for λ n in terms of fundamental units where n is a fundamental discriminant of an imaginary quadratic field Q n which has only one class in each genus of ideal classes In particular he calculated λ 7 λ 4 λ 65 λ 89 and λ 65 This

6 70 BERNDT CHAN KANG AND ZHANG formula and all 4 values of such λ n s are given in Section 6 In the same section we extend Ramanathan s method to establish a similar result for λ n when n mod 4 and there is precisely one class per genus in each imaginary quadratic field Q n Through Section 6 all values of λ n in are calculated except for n = In Section 7 we employ an empirical process analogous to that employed by G N Watson [9] [0] in his calculations of class invariants to determine λ n for these remaining values of n This empirical method has been put on a firm foundation by Chan A Gee and V Tan [4] Their method works whenever n n is squarefree and the class group of Q n takes the form Z Z Z k with k 4 The first representation of λ n in 4 suggests connections between λ n and Ramanujan s alternative cubic theory In fact Berndt and Chan [5] have recently found such a relationship In Section 8 applying one of their results we establish an explicit formula for the j-invariant in terms of λ n and evaluate several values of the j-invariant Values of the j-invariant play an important role in generating rapidly convergent series for /π For example using the value of λ 05 Berndt and Chan [6] established a series for /π which yields about 7 or 74 digits of π per term The previous record which yields 50 digits per term was given by the Borweins [9] in 988 The values of λ n are not only related to convergent series for /π via the j-invariants In fact Chan Liaw and Tan generated [6] a new class of series for /π depending on values of λ n Using the values of λ n proved in this paper they derived many simple series for /π which are analogues of Ramanujan s series belonging to the theory of q For example he proved that 4 π = 5k k=0 k k k k 9 6 k which follows from the value λ 9 = and a certain Lambert-type series identity In the table we observe that if n is not divisible by then λ n is a unit In fact in the final section we show that λ n is a unit when n is odd and n We conclude the introduction by summarizing in a table the values of n for which λ n is determined in this paper and the sections where the values can be located

7 QUOTIENT OF ETA-FUNCTIONS 7 n Sections n Section n Section n Section n Section / / / / / λ n and the modular j-invariant Recall that [7 p 8] the invariants Jτ and jτ for τ H := {τ : Im τ > 0} are defined by Jτ = g τ τ where and jτ = 78Jτ τ = gτ 7gτ g τ = 60 mτ n 4 and g τ = 40 mn= mn 00 mn= mn 00 mτ n 6 Furthermore the function γ τ is defined by [8 p 49] γ τ = jτ

8 7 BERNDT CHAN KANG AND ZHANG where that branch which is real when τ is purely imaginary is chosen In his third notebook at the top of page 9 in the pagination of [6] Ramanujan defines a function J n by 4 J n = n γ = n j For 5 values of n n mod 4 Ramanujan indicates the corresponding values for J n See [ pp 0-] for proofs of these evaluations In particular J = 0 J 7 = 5 / J 5 = 7 4 / J 75 = 5 / J 99 = 4 / 77 5 The first five values of n for λ n in are those for which n = ; the corresponding values of J n are given in 5 Then on the next page which is the last page of his third notebook Ramanujan gives a formula leading to a representation of λ n Theorem Ramanujan For q = exp π n define 6 R := R n := /4 /6 fq q fq / Then 7 R 6 n = 8J n 64Jn 4J n 9 8J n 6 Theorem was first proved in [ p 8 Entry ] Since λ n = Rn 6 by 4 and 6 7 may be restated as 8 λ n/ = 8J n 64Jn 4J n 9 8J n 6 By substituting J = 0 into 8 we determine the first value of λ n in and we state it as a corollary Corollary λ = Unfortunately it is not so easy to find other values of λ n from Theorem We have to struggle with complicated radicals even when n = 9 for which λ 9 = It seems that Ramanujan used this formula to determine the values of λ n/ for rational integral values of J n as given in the following table which constitutes the first part of the last page of the third notebook

9 QUOTIENT OF ETA-FUNCTIONS 7 9 n J n 8J n 64Jn 4J n 9 49 = = = = = = = = But we are going to use Ramanujan s discoveries recorded between the table 9 and Theorem on the last page in his third notebook Ramanujan first sets for q = exp π n 0 t n := q /8 fq/ fq f q and u n := 8 J n To avoid a conflict of notation we have replaced Ramanujan s second t n by u n He then asserted that t n = /6 64Jn 4J n 9 6J n and listed very simple polynomials satisfied by t n and u n The definition of u n in seems unmotivated but by recalling from the proof of Theorem in [ p ] that 8J n = f 6 q / q /6 f 6 q 7 q f 6 q f 6 q we find that 4 u n = f 6 q / q /6 f 6 q q f 6 q f 6 q We summarize these results in the following two corollaries Corollary λ n/ λ n = 8J n Proof This is a restatement of either or 4 with the definition of λ n in 4

10 74 BERNDT CHAN KANG AND ZHANG Corollary 4 λ n λ n/ = 64J n 4J n 9 6J n Proof By 4 and 0 t 6 n = 7λ n/ λ n We obtain the result at once from Corollary 5 λ 9 = Proof Let n = in either Corollary or Corollary 4 The result follows immediately from the fact that J = 0 and λ = Corollary 6 i ii λ / = λ = = 8 Proof i is an immediate consequence of Theorem with n = since J = Using i and either Corollary or Corollary 4 when n = we obtain ii Corollary 7 λ 9/ = / i 57 λ 57 = / ii 57 Proof Using 8 with n = 9 we find that λ 9/ = /4 9 Let us represent x = 9 as a product of units If t = 9 then x t = or 5 x tx 9 5 = 0 Let y = x 9 5 Then 5 becomes 6 x 9 5 t y = y

11 QUOTIENT OF ETA-FUNCTIONS 75 Hence by applying the quadratic formula to y y = 9 = we find that y = from which i follows From i and either Corollary or Corollary 4 with n = 9 we deduce ii By similar methods we can derive the following results Corollary 8 i λ 4/ = / ii λ 9 = / Corollary i ii λ 67/ = / λ 0 = / Corollary 0 i λ 6/ = 486 / ii λ 489 = 486 /

12 76 BERNDT CHAN KANG AND ZHANG λ n and the class invariant G n In [] Ramanathan introduced a new function µ n defined by µ n := ηi 6 n/ ηi = n f 6 q qf 6 q 6 q = e π n/ Then by 4 and Euler s pentagonal number theorem f q = q; q λ n fqf q = q / 6 6 q; q µ n f q fq = q / 6 q ; q 6 Hence from 7 we deduce the following result: Theorem Let P = f q q / f q λ n µ n = Gn/ G n 6 and Q p = f q p q p/ f q p Recall the modular equation [ p 04 Entry 5] P Q 9 6 P Q = Q P 6 P By replacing q by q in we deduce from 4 and Theorem that λ n µ n / λ n µ n / Gn Gn/ = G n/ Solving 4 for λ n µ n we find that c c 5 λ n µ n = 9 6 where c = Gn Gn/ 6 G n/ G n Hence from Theorem and 5 we derive the following theorem Theorem λ n = Gn/ G n where c = Gn G n/ 6 Gn/ G n 6 Q c c 9 G n

13 QUOTIENT OF ETA-FUNCTIONS 77 We give another proof of Corollary Corollary i ii λ = µ = Proof Since G /n = G n [4] G / /G 6 = Substituting this value into Theorem we have i and then using Theorem we deduce ii at once Corollary 4 λ = /4 Proof Let n = in Theorem and use the values [ p 89] / G = and G 9 = 4 λ n and modular equations We shall employ a certain type of modular equation of degree p in P and Q p defined in to calculate several values of λ n First recall the modular equation of degree 9 [ p 46 Entry iv] 4 9q f q 9 f q = 7q f q f q / After replacing q by q in both sides we deduce the following result from the definition of λ n in 4 Theorem 4 Corollary 4 i ii λ n = λ n λ 9n λ 9 = λ 8 = Proof Let n = and n = 9 in Theorem 4 to obtain i and ii respectively At the end of Section we remarked that λ n is a unit for odd n not divisible by With this as motivation set 4 λ n = a a 6

14 78 BERNDT CHAN KANG AND ZHANG and let 4 Λ = λ / n λ / n Then 44 a = Λ 4 By determining Λ in 4 and then using 44 and 4 we next use modular equations to prove all the evaluations of λ p given explicitly by Ramanujan in Theorem 4 7λ n λ 5n / 7 / = λ n λ 5n λ5n λ n / / λn 5 λ 5n Proof From [ p Entry 6] we find that 45 P Q P Q 5 = Q5 P P where P and Q 5 are defined by We can deduce Theorem 4 from 45 and 4 immediately after replacing q by q Corollary 44 λ 5 = 6 5 = 5 Proof For brevity we set λ = λ 5 in the proof Let n = in Theorem 4 Then we have 46 λ / λ / = λ / λ / 5 Set Λ = λ / λ / Since 46 becomes Q 5 λ / λ / = λ /6 λ /6 λ / λ / Λ 5 = Λ / Λ which can be simplified after squaring both sides to Λ = 0 Thus Λ = and a = /4 by 44 Hence from λ 5 = 4 4

15 QUOTIENT OF ETA-FUNCTIONS 79 Theorem 45 / / 7 7λ n λ 49n λ n λ 49n / / λ49n λ49n = 7 7 λ n λ n λn λ 49n / / λn λ 49n Proof By and 4 this theorem can be deduced from [ p 6 Entry 69] P Q 7 7 P Q 7 = with q replaced by q Corollary 46 λ 49 = Q7 P 4 7 Q7 6 7 = 55 Proof Let n = in Theorem 45 and set λ = λ 49 Then P P P 4 7 Q 7 Q 7 47 λ / λ / = λ / 7λ / 7λ / λ / = λ / λ / λ / λ / 7 which can be simplified to 48 λ / λ / = λ /6 λ /6 λ / λ / 7 Letting Λ = λ / λ / in 48 we have 49 Λ = Λ Λ 7 Squaring both sides of 49 we deduce that Hence Λ = 5 and by 4 and 44 Λ 5 Λ = 0 λ 49 =

16 80 BERNDT CHAN KANG AND ZHANG Theorem 47 9 {λ n λ n 5/6 λ n λ n 5/6 } 99{λ n λ n / λ n λ n / } 98 {λ n λ n / λ n λ n / } 759{λ n λ n / λ n λ n / } 69 {λ n λ n /6 λ n λ n /6 } 86 λn λn = λ n λ n Proof A new modular equation of degree which was not mentioned by Ramanujan was proved by Berndt Bhargava and Garvan [4] and is given by 40 { 5 } { 4 P Q 5 P Q 4 66 P Q P Q P Q } { } { 5 P Q 69 P Q P Q P Q } P 6 6 Q 68 = P Q Q P where P and Q are defined by Replacing q by q in the equation above yields Theorem 47 Corollary 48 λ = = Proof Letting n = and Λ = λ / λ / in Theorem 47 we deduce that 4 9 Λ Λ Λ / 99Λ 98 Λ Λ / 759Λ 69 Λ / 86 = ΛΛ Rearranging 4 to 9 Λ / Λ Λ 54 = Λ 99Λ 756Λ 88

17 QUOTIENT OF ETA-FUNCTIONS 8 and then squaring both sides we deduce the equation Thus Λ 5Λ 8 = 0 Λ = 5 We complete the proof by using 4 and 44 We now establish λ 5 λ 7 and λ as well by using Theorems and the following lemma Lemma 49 Proof Recall that λ /n = λ n 4 ητ = e πi/ ητ and η = i / τ / ητ τ From these properties of the Dedekind eta-function or Entry 7iv Ch 6 in [ p 4] we find that η 6 i/ n = n n η 6 i n and η 6 i /n = n Hence the lemma follows from 4 Corollary 40 n η 6 λ 5 = 5 i n/ Proof Letting n = /5 in Theorem 4 and using Lemma 49 we have = λ 5 λ 5 Solving this equation for λ 5 completes the proof Corollary 4 λ 7 = / 7 /

18 8 BERNDT CHAN KANG AND ZHANG Proof Letting n = /7 in Theorem 45 yields by Lemma = λ 4/ 7 7λ / 7 7λ / 7 λ 4/ 7 If we set x = λ / 7 λ / 7 then 4 is equivalent to 6 = x x 4 7x which has the solution x = Hence by the quadratic formula λ / 7 = 7 7 = 7 = 7 = 7 Then the value of λ 7 follows immediately Corollary 4 λ = / 0 / Proof If n = / in Theorem 47 then by Lemma 49 λ λ = By the quadratic formula λ = = = = 0 Taking the positive square root above we obtain the value of λ 5 λ n and modular equations in the theory of signature Suppose β has degree p over α in the theory of signature and let z := F ; ; α and z p := F ; ; β For ω = expπi/ if the cubic theta functions are defined by 5 aq := q m mnn 5 bq := mn= mn= ω m n q m mnn

19 QUOTIENT OF ETA-FUNCTIONS 8 and 5 cq := mn= then [4 Lemma 6 and Corollary ] q m/ m/n/n/ aq = z bq = α / z cq = α / z and aq p = z p bq p = β / z p cq p = β / z p Thus it follows that cqcq 54 x := αβ /6 p / = aqaq p and 55 y := { α β} /6 = But since [4 Lemma 5] 56 bq = f q f q 57 cq = q / f q f q and [0] 58 a q = b q c q we find that 59 aq = { bqbq p / aqaq p } / f q 7qf q f qf q This was also proved by Ramanujan; see [ p 460 Entry i] Hence by and x = and 5 y = q p/6 f q f q p {f q 7qf q } /6 {f q p 7q p f q p } /6 f qf q p {f q 7qf q } /6 {f q p 7q p f q p } /6 Let 5 T := T q := f q /4 q / f q and U p := U p q := f q p /4 q p/ f q p

20 84 BERNDT CHAN KANG AND ZHANG Then from 50 and 5 we find that 5 and x y = T U p 54 xy = T 6 T 6 / U 6 p U 6 p / We now employ modular equations in x and y to calculate further values of λ n Theorem 5 { λ n λ 89n / λ n λ 89n /} 80 {λ n λ 89n / λ n λ 89n /} 8 λ n λ n / λ 89n λ 89n / λ n λ 89n / λ n λ 89n / { 5 λ n λ 89n / λ n λ 89n /} 56 λ n λ n / λ 89n λ 89n / λ n λ 89n / λ n λ 89n / 4 = { λn λ 89n } λn λ 89n Proof The modular equation of degree 7 with which we start was proved by Chan and W-C Liaw [5] and is given by 55 x 6 96x 5 y 40x 4 y 54x y 40x y 4 96xy 5 y 6 x 4 05x y 68x y 05xy y 4 x 4xy y = 0 where x and y are defined by 54 and 55 with p = 7 Dividing both sides of 55 by x y we obtain x y y x x 80 y y 56 8 x { x xy y y x x 5 y y } 56 x { x x y y y } 4 = { } x x x y y y x after a slight rearrangement

21 QUOTIENT OF ETA-FUNCTIONS 85 By 5 54 and {T U 7 4 T U 7 4 } 80{T U 7 T U 7 } 8 T 6 T 6 / U7 6 U7 T 6 / U 7 4 T U {T U 7 T U 7 } 56 T 6 T 6 / U7 6 U7 T 6 / U 7 T U 7 4 { = T 6 } T 6 U 7 U 7 Replacing q by q in 57 setting q = e π n/ and using 4 we complete the proof Corollary λ 89 = Proof Set n = and λ = λ 89 It follows from Theorem 5 that 58 8 λ / λ / 80λ / λ / = λ λ Let Λ := λ / λ / Then from 58 we have 8 Λ 80Λ = Λ Λ Hence Λ is a root of the equation The proper solution for Λ is Λ 96Λ 4Λ 6 = 0 Λ = Corollary 5 now follows from 4 and 44 Corollary 5 λ 7 = 4 7 Proof Let n = /7 in Theorem 5 Then by Lemma 49 we have 7 64λ 7 λ 7 / 6λ 7 λ 7 4/ = 6 λ 7 λ 7 We complete the proof by solving this equation for λ 7

22 86 BERNDT CHAN KANG AND ZHANG Theorem 54 λ 69 = 6 4 Proof Let x and y be defined by 54 and 55 with p = In [0] W-C Liaw established a modular equation of degree which by 5 and 54 can be put in the abbreviated form { T U 4 T U 4} 764 { T U 6 T U 6} { T U 0 T U 0} { T U 4 T U 4} { T U 8 T U 8} { T U T U } { T U 6 T U 6} Rx y = 0 where Rx y contains a factor /x y Let q = e π/ recall that λ = and set λ = λ 69 Replacing q by q in 59 by 4 we find that 50 λ 7 λ 7 764λ 6 λ λ 5 λ λ 4 λ λ λ λ λ λ λ = 0 since Rx y equals 0 after q is replaced by q for /x y = T 6 T 6 U 6 U 6 is a factor of Rx y and { T 6 q T 6 q } { U 6 q U 6 q} = { λ λ λ 69 λ 69 } Set Λ = λ λ Then 50 has the equivalent form Λ 7 7Λ 5 4Λ 7Λ 764Λ 6 6Λ 4 9Λ Λ 5 5Λ 5Λ Λ 4 4Λ Λ Λ Λ Λ = 0 which simplifies to Therefore Λ Λ 580Λ 900 = 0 λ λ = Λ = 069 5

23 QUOTIENT OF ETA-FUNCTIONS 87 Solving the quadratic equation above we find that λ 69 = Using the formula [7 Eq ] { b 6b } /6 b 6b = b b with b = 5 4 we deduce the value of λ 69 4 Equation 59 can also be used to deduce λ Theorem 55 λ 6 = / 4969 / / / Proof The proof is similar to that for Theorem 54 Let x and y be defined by 54 and 55 respectively and set u = x v = y and p = 9 Liaw [0] found a modular equation of degree 9 which we give in the abbreviated form u 0 v 0 { u 8 v 8 v 0 u 0 v 8 u { u 9 v 9 v 9 u 9 } { u v 6 { u 5 v 5 v 5 u 5 { u v { u v v u v 6 u 6 } v u } Ru v = 0 } { u 7 v 7 } v 7 u 7 } { u 4 v 4 } v 4 u 4 } { u v v u} where Ru v is a sum of terms with a factor /uv which equals 0 after setting q = e π/ and replacing q by q With Λ = λ 6 λ 6 we

24 88 BERNDT CHAN KANG AND ZHANG eventually find that Λ 0 0Λ 8 5Λ 6 50Λ 4 5Λ Λ 9 9Λ 7 7Λ 5 0λ 9Λ Λ 8 8Λ 6 0Λ 4 6Λ Λ 7 7Λ 5 4λ 7Λ Λ 6 6Λ 4 9Λ Λ 5 5Λ 5Λ Λ 4 4Λ Λ Λ Λ Λ = 0 which simplifies to Λ Λ 58576Λ Λ = 0 Hence Λ = / 8599 / Solving the equation Λ = λ 6 λ 6 for λ 6 we complete the proof The modular equation of degree 9 given above can also be used to calculate λ 9 ; see also Corollary 65iii 6 λ n and Kronecker s limit formula Let m > 0 be square-free and let K = Q m the imaginary quadratic field with discriminant d where { 4m if m mod 4 6 d = m if m mod 4 Let d = d d where d > 0 and for i = d i 0 or mod 4 If P denotes a prime ideal in K then the Gauss genus character χ is defined by d if NP d NP 6 χp = d if NP d NP where NP is the norm of the ideal P and denotes the Kronecker symbol

25 QUOTIENT OF ETA-FUNCTIONS 89 Let m if m mod 4 6 Ω = m if m mod 4 It is known [] that each ideal class in the class group C K contains primitive ideals which are Z-modules of the form A = [a b Ω] where a and b are rational integers a > 0 a Nb Ω b a/ a is the smallest positive integer in A and NA = a Hence Siegel s Theorem [7 p 7] obtained from Kronecker s limit formula can be stated as follows Theorem 6 Siegel Let χ be a genus character arising from the decomposition d = d d Let h i be the class number of the field Q d i ω and ω be the numbers of roots of unity in K and Q d respectively and ɛ be the fundamental unit of Q d Let 64 F A = ηz a where z = b Ω/a with [a b Ω] A Then 65 ɛ ωh h /ω = F A χa A C K Ramanathan utilized Theorem 6 to compute λ n and µ n [ Theorem 4] Theorem 6 Ramanathan Let n be a positive square free integer and let K = Q n be an imaginary quadratic field such that each genus contains only one ideal class Then where χ ɛ tχ = { λn if n mod 4 µ n if n mod 4 66 t χ = 6ωh h ω h h h h are the class numbers of K Q d and Q d respectively ω and ω are the numbers of roots of unity in K and Q d respectively ɛ is the fundamental unit in Q d and χ runs through all genus characters such that if χ corresponds to the decomposition d d then either d or d = and therefore d d h h ω and ɛ are dependent on χ By Theorem 6 4 values of λ n and 9 of µ n can be evaluated Among them Ramanujan recorded only the values of λ n for which the exponent t χ = We state all 4 values of such λ n s in the following corollary

26 90 BERNDT CHAN KANG AND ZHANG Corollary 6 λ 5 = 5 λ 7 = 4 7 λ 4 = 5 4 λ 65 = λ 89 = λ 45 = λ 6 = λ 85 = λ 09 = λ 65 = λ 85 = λ 665 = λ 00 = λ 05 = We next derive a theorem from Theorem 6 by which we can evaluate some λ n for n mod 4 Theorem 64 Let n > be a positive square free integer not divisible by and n mod 4 Let K = Q n be an imaginary quadratic field such that each genus contains only one ideal class Let C 0 be the principal ideal class containing [ Ω] where Ω is defined by 6 and C and C be nonprincipal ideal classes containing [ Ω] and [6 Ω] respectively

27 QUOTIENT OF ETA-FUNCTIONS 9 Then λ n = χc = ɛ tχ χc = where t χ d d h h ω and ɛ are defined in Theorem 6 and the products are over all characters χ the first with χc = and the second with χc = associated with the decomposition d = d d and therefore d d h h ω and ɛ are dependent on χ Proof If A C is any of the ideals [ Ω] [ Ω] and [6 Ω] then A A C = C and A C For any ideal class C not C 0 nor C Ramanathan [] showed that χc = 0 which implies that χc = χc = F C χc = ɛ tχ where F C is defined by 64 Therefore by 65 ɛ ωh h /ω = 67 F A χa A C 0 C χc = χc = = A C 0 C F A χah/ = F C h/ F C 0 since the number of genus characters is h and so the number of genus characters such that χc = is h/ With a similar argument we find that 68 χc = Dividing 67 into 68 we have ɛ ωh h /ω = F C h/ F C 0 69 But by F C F C h/ = χc = ɛωh h /ω F C F C = χc = ɛωh h /ω η η i n/ i n The theorem now follows from and 4

28 9 BERNDT CHAN KANG AND ZHANG We can apply Theorem 64 to evaluate λ n when n = We first determine all the values of such λ n when Q n has class number 4 and each genus contains one ideal class Observe that χc = where χ has the decomposition d = d d such that d = or d = ie d ± or d ± mod 8 and χc = where d 6 = or d 6 = Corollary 65 i ii iii iv v λ 7 = / 7 / λ = / 0 / λ 9 = /4 / λ = 9/ 7 / λ 59 = 0 59 / / Proof For all five values of n above the corresponding imaginary quadratic field Q n has class number 4 and genus number 4 Since the values of λ 7 and λ are already determined in Corollaries 4 and 4 respectively we will discuss only the last three values The following tables summarize the needed information about ideal classes and their characters Let K = Q 57 d d χ C 8 χ 0 [ Ω] 57 4 χ [ Ω] 44 χ [6 Ω] χ [ Ω] By Theorem 64 χ λ 9 = χ ɛ tχ χ χ ɛ tχ Let K = Q 9 = ɛtχ ɛ tχ χc 0 χc χc χc h h ω ɛ = /4 /

29 QUOTIENT OF ETA-FUNCTIONS 9 d d χ C 7 χ 0 [ Ω] 9 4 χ [ Ω] 44 χ [6 Ω] χ [ Ω] χc 0 χc χc χc h h ω ɛ By Theorem 64 λ = χ χ ɛ tχ χ χ ɛ tχ = ɛtχ ɛ tχ = 9/ 9 /4 9 Let K = Q 77 d d χ C 708 χ 0 [ Ω] 77 4 χ [ Ω] 6 χ [6 Ω] 59 χ [ Ω] χc 0 χc χc χc h h ω ɛ By Theorem 64 χ λ 59 = χ ɛ tχ χ χ ɛ tχ = ɛtχ ɛ tχ = / / = 0 59 / / Next we give examples of evaluations of λ n when Q n has class number 8 and each genus contains one ideal class Corollary 66 i λ 5 = 5 / 4 5 / 5 5 7

30 94 BERNDT CHAN KANG AND ZHANG ii iii iv λ 55 = 0 / 5 /4 65 λ 9 = / /4 / 5 / λ 5 = 4 5 / / 5 / v λ 9 = / 0 9 / 9 /4 57 Proof We give the calculation for only i as the calculations in the remaining four cases are similar Let K = Q 05 Then its discriminant d = 40 χc 0 χc d d χ C h χc χc h ω ɛ 40 χ 0 [ Ω] 05 4 χ [ Ω] χ [6 Ω] χ [ Ω] 5 84 χ 4 0 χ χ χ 7 By Theorem

31 QUOTIENT OF ETA-FUNCTIONS 95 λ 5 = χ χ χ 5 χ 6 ɛ tχ χ χ χ 4 χ 5 ɛ tχ = ɛtχ ɛ tχ 6 ɛ tχ ɛ tχ 4 = /4 4 5 / 6 / 5 5 Lastly by a similar calculation we can determine λ 455 ; here Q 65 has class number 6 and each genus contains one ideal class Corollary 67 λ 455 = 4 95 / / / / 7 An empirical process for computing λ n 7 /4 65 In this section we describe an empirical process which can be used to derive values of λ n for n = and 4 This empirical process is analogous to those used by GN Watson [9] [0] in his computations of the Ramanujan Weber class invariants G n and g n Theorem 7 We have i ii iii iv v λ 7 = λ 97 = λ 4 = λ / 9 λ / 9 λ 7 = = / /

32 96 BERNDT CHAN KANG AND ZHANG Proof By computing the numerical value of λ 7 using we assume that λ / 7 λ / = = Solving this quadratic equation we arrive at λ 7 = Verifying that the value above is the same as Ramanujan s value for λ 7 is straightforward The computations of λ 97 and λ 4 are similar We assume that and which yield and λ / 97 λ / 97 λ / 4 λ / 4 λ 97 = λ 4 = = = 97 = = The computation of λ 7 is different from those of the previous three values We compute λ 7 and λ /7 numerically and assume that and λ 7 λ /7 / λ7 λ /7 / which imply that λ 7 = λ 7 λ /7 / = = 7 7 λ/7 λ / = = / /

33 QUOTIENT OF ETA-FUNCTIONS 97 In [4] this empirical method will be put on a firm foundation and it will be shown that λ n can be explicitly determined when n mod 4 n and the class group of Q n is of the type Z Z Z Z 4 This result is analogous to that associated with the Ramanujan-Weber class invariant G n proved in [] It is clear that this method works for groups of the type Z Z Z which is the class group of imaginary quadratic fields having one class per genus Hence the empirical process illustrated here can also be used to determine the values of λ n given in Section 6 It remains to determine the value of λ 9 The class group of Q 579 is Z 8 and so it does not belong to the set of n s that we discuss here However we know that [4] t := λ 9 λ 9 lies in a quartic extension of Q 579 and so we may use MAPLE V s minpoly function to guess a minimal polynomial of degree 4 satisfied by t Our final result is λ / 9 λ / 9 = 4 A rigorous proof of 7 can be found in [4] Evaluation of the modular j-invariant The base q in the alternative cubic theory is defined by 8 q := q := exp π F ; ; α F 0 < α < ; ; α where F a b; c; z is defined by 5 Recall the fundamental inversion formula for the cubic theta functions [4 Lemma 9] 8 α := αq = c q a q where the cubic theta functions aq bq and cq are defined by 5 5 By we find that 8 Let αq = f q 7qf q 84 s := sq := f q 7qf q From the relation between the j-invariant and the modulus α in the cubic theory [5] 85 jτ = 7 8 α αα q = e πiτ

34 98 BERNDT CHAN KANG AND ZHANG and 8 we deduce that 86 jτ = 79s s s Thus by 87 γ τ = s jτ = 9s s Let τ = n/6 Then by 4 and 4 we obtain the following theorem after replacing n by n and dividing both sides by Theorem 8 J n = γ n = 9λ n λ n λ n By Theorem 8 we can determine the corresponding values of J n for known values of λ n For example the values of J n in the table 5 are easy consequences of Theorem 8 We give another example in the corollary below Corollary 8 J = Proof Let n = 4 in Theorem 8 and use the value of λ 4 in Table 9 λ n is a unit when n is odd and not divisible by Recall that an order O f with conductor f in a quadratic field K is a subset O f K such that i O f is a subring of K containing ii O f is a finitely generated Z-module iii O f contains a Q-basis of K and iv [O K : O f ] = f where O K = O is the ring of integers of K If α and α generate an O f -ideal A Of over Z we say that [α α ] is a basis of A Of An O K ideal A Of is said to be proper if O f = {β K : βa Of A Of } Every ideal in O K prime to f is proper If ν is an algebraic integer and A is an O K ideal we write ν A to mean that νo L = AO L in some larger number field L Similarly if ν and ν are algebraic integers we write ν ν to mean that ν /ν is a unit

35 QUOTIENT OF ETA-FUNCTIONS 99 Theorem 9 Deuring [9 p 4] Let f be a positive integer and p t f where p is a prime and t is a nonnegative integer Let a b c and d be integers M := be a matrix with determinant p and τ = qq; q a b c d 4 where q = e πiτ with Im τ > 0 Define 9 Φ M τ = p M τ τ τ τ where τ = τ /τ and τ τ := τ τ := τ τ Let [α α ] be a basis of a proper O f -ideal A Of and set α = α /α The action of M on the basis [α α ] is defined as M[α α ] = [aα bα cα dα ] I When p splits completely in K namely p = PP then Ia Φ M α is a unit if M[α α ] is a basis of a proper O fp -ideal Ib Φ M α p if M[α α ] is a basis of a proper O fp -ideal Ic if p f then Φ M α P and Φ M α P when M[α α ] is a basis of A Of P Of and M[α α ] is a basis of A Of P Of respectively II When p ramifies in K namely p = P then IIa Φ M α p 6/pt if M[α α ] is a basis of a proper O fp -ideal IIb Φ M α p 6/pt if M[α α ] is a basis of a proper O fp -ideal IIc Φ M α p 6 if M[α α ] is a basis of A Of P Of III When p is inert in K then IIIa Φ M α p /pt p if M[α α ] is a basis of a proper O fp -ideal IIIb Φ M α p /pt p if M[α α ] is a basis of a proper O fp - ideal Corollary 9 λ n is a unit if n is odd and not divisible by Proof Let n = f d where d is square-free and let K = Q d i First suppose that n mod 4 Then d mod 4 and the ring of integers O K = [ d ] and = P ramifies in [ O K Consider ] the f f order O f of conductor f Then O f has a basis O f = d Let [ ] [ ] f f f M = Then M d f f 0 = d is a basis of PO f Hence by IIc in Theorem 9 we have 9 Φ M f f d 6

36 00 BERNDT CHAN KANG AND ZHANG On the other hand f f f f Φ f d d M = 0 f f d = f f d 6 f f d 6 f f d = 6 f f d f f d/ n/ = = n f f d where the last equality follows from the facts that τ = τ and f is odd But since τ = η 4 τ by f 4 i n/ f Φ d η M = η i n Thus by 4 f f 9 Φ d M = λ n 4 = 6 λ 4 n Combining 9 and 9 we have 6 6 λ 4 n Hence λ 4 n is a unit and so is λ n ii Suppose that n mod 4 Then d mod 4 O K = [ d ] and = P ramifies in O K The order O f of conductor f has a basis O f = [ f d ] Let [f f d ] be a basis of a proper O f -ideal f A Of Let M = Since M[f f 0 d ] = [f f d 6] is a basis of A Of P Of we deduce from IIc in Theorem 9 that f f Φ d M 6 As we have shown above Φ M f f d = 6 λ 4 n Hence we can conclude that λ n is a unit

37 QUOTIENT OF ETA-FUNCTIONS 0 Corollary 9 If n is odd and congruent to modulo then / λ n is a unit Proof Let n = f d where d is square-free and let K = Q d [ ] i Suppose that n mod 4 Then d mod 4 O K = d and = PP splits completely in O K because d [ mod Let ] O f f f be an order of O K of conductor f Then O f = d Let [ ] [ ] 0 f f M = Then M d f f 0 = d is a basis of O f Hence by Ib of Theorem 9 f f 94 Φ d M On the other hand by a calculation like that in the previous proof f 0 f f Φ f d d M = 0 95 f f d = λ n 4 = 6 λ 4 n by and 4 Combining 94 and 95 we deduce that / λ n is a unit ii Suppose that n mod 4 Then d mod 4 O K = [ d ] and = PP splits completely in O K Let O f = [ f d ] be an order of O K with conductor f Let [f f d ] be a basis of a proper O f - ideal A f Let M = f 0 Then M[f f d ] = [f f d 6] is a basis of B Of where B Of = [f f d ] is a proper ideal of O f Hence by Ib in Theorem 9 f f 96 Φ d M On the other hand by calculations like those above f f f f 97 Φ f d d M = 0 f f d = 6 λ 4 n where the last equality was deduced from and 4 Hence from 96 and 97 we can conclude that / λ n is a unit

38 0 BERNDT CHAN KANG AND ZHANG Corollary 94 If n is odd and congruent to modulo then /4 λ n is a unit Proof Let n = f d where d is square-free and let K = Q d i Suppose that n mod 4 Then d mod 4 O k = [ d ] and is inert in O K because d mod By an argument similar to that in i in the proof of Corollary 9 and by the use of IIIb in Theorem 9 we can deduce that 6 λ 4 n = Φ M Hence /4 λ n is a unit f f d /4 = 9 ii We can deduce the same result when n mod 4 by using a method similar to that in ii in the proof of Corollary 9 and by using IIIb of Theorem 9 Corollary 95 If n = t for t then t λ n is a unit Proof Suppose that n mod 4 Let K = Q Then O K = [ ] and [ t t ] is a basis of the order O t with conductor t 0 Note that =P ramifies in O K Let M = Then M[ 0 t t ] = [ t t ] is a basis of the principal ideal O t By IIb in Theorem 9 t Φ t M 6 t But since by and 4 t Φ t M = 6 λ 4 n t λ n is a unit We are very grateful to W-C Liaw for providing the necessary modular equations of degrees and 9 We also thank Jinhee Yi for discovering several calculational errors in an earlier draft of this paper References [] BC Berndt Ramanujan s Notebooks Part III Springer-Verlag New York 99 MR 9j:0069 Zbl 700 [] Ramanujan s Notebooks Part IV Springer-Verlag New York 994 MR 95e:08 Zbl 78500

39 QUOTIENT OF ETA-FUNCTIONS 0 [] Ramanujan s Notebooks Part V Springer-Verlag New York 998 MR 99f:04 Zbl [4] BC Berndt S Bhargava and FG Garvan Ramanujan s theories of elliptic functions to alternative bases Trans Amer Math Soc MR 97h:04 Zbl 840 [5] BC Berndt and HH Chan Ramanujan and the modular j-invariant Canad Math Bull CMP [6] Eisenstein series and approximations to π Illinois J Math [7] BC Berndt HH Chan and L-C Zhang Radicals and units in Ramanujan s work Acta Arith MR 99m:5 Zbl [8] JM Borwein and PB Borwein Pi and the AGM Wiley New York 987 MR 89a:4 Zbl 6000 [9] More Ramanujan-type series for /π in Ramanujan Revisited eds GE Andrews RA Askey BC Berndt KG Ramanathan and RA Rankin Academic Press Boston MR 89d:8 Zbl [0] A cubic counterpart of Jacobi s identity and the AGM Trans Amer Math Soc MR 9e:0 Zbl 7504 [] Class number three Ramanujan type series for /π J Comput Appl Math MR 94h: Zbl [] HH Chan On Ramanujan s cubic transformation formula for F / /; ; z Math Proc Cambridge Philos Soc MR 99f:054 Zbl [] Ramanujan-Weber class invariant G n and Watson s empirical process J London Math Soc MR 99j:4 Zbl 97 [4] HH Chan A Gee and V Tan Cubic singular moduli Ramanujan s class invariants λ n and the explicit Shimura reciprocity law Pacific J Math to appear [5] HH Chan and W-C Liaw On Russell-type modular equations Pacific J Math [6] HH Chan W-C Liaw and V Tan Ramanujan s class invariant λ n and a new class of series for /π J London Math Soc [7] K Chandrasekharan Elliptic Functions Springer-Verlag Berlin 985 MR 87e:058 Zbl [8] DA Cox Primes of the Form x ny Wiley New York 989 MR 90m:06 Zbl 7000 [9] M Deuring Die Klassenkörper der komplexes Multiplication in Enz Math Wiss Band I Heft 0 Teil II Stuttgart 958 [0] W-C Liaw Contributions to Ramanujan s Theory of Modular Equations in Classical and Alternative Bases PhD Thesis University of Illinois 999 [] R Mollin and L-C Zhang Orders in quadratic fields II Proc Japan Acad Ser A MR 95d:5 Zbl [] KG Ramanathan Some applications of Kronecker s limit formula J Indian Math Soc MR 90j: Zbl 6800 [] On some theorems stated by Ramanujan in Number Theory and Related Topics Tata Institute of Fundamental Research Studies in Mathematics Oxford University Press Bombay MR 98g:07 Zbl

40 04 BERNDT CHAN KANG AND ZHANG [4] S Ramanujan Modular equations and approximations to π Quart J Math Oxford [5] Notebooks volumes Tata Institute of Fundamental Research Bombay 957 MR 0 #640 Zbl 840 [6] The Lost Notebook and Other Unpublished Papers Narosa New Delhi 988 Zbl 6900 [7] CL Siegel Advanced Analytic Number Theory Tata Institute of Fundamental Research Bombay 980 MR 8m:000 Zbl [8] H Stark Values of L-functions at s = i L-functions for quadratic forms Advances in Math MR 44 #660 Zbl 6005 [9] GN Watson Theorems stated by Ramanujan XIV: A singular modulus J London Math Soc [0] Some singular moduli I Quart J Math [] H Weber Lehrbuch der Algebra dritter Band Chelsea New York 96 Received February 000 and revised June Department of Mathematics University of Illinois Urbana IL 680 address: berndt@mathuiucedu Department of Mathematics National University of Singapore Kent Ridge Singapore 960 Singapore address: chanhh@mathnusedusg Department of Mathematics The Ohio State University Columbus OH 40 address: kang@mathohio-stateedu Department of Mathematics Southwest Missouri State University Springfield Missouri address: liz97f@smsuedu