Arithmetic Properties for Ramanujan s φ function

Transcription

1 Arithmetic Properties for Ramanujan s φ function Ernest X.W. Xia Jiangsu University ernestxwxia@163.com Nankai University Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

2 Outline 1. Introduction Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

3 Outline 1. Introduction 2. Congruences modulo powers of 2 (proved by theta function identities) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

4 Outline 1. Introduction 2. Congruences modulo powers of 2 (proved by theta function identities) 3. Congruences modulo powers of 3, 5 and 7 (proved by Newman s identities) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

5 Identities involving φ(q) Let p(n) denote the number of partitions of n. Owe to Ramanujan, it is well-known that for n 0, p(5n + 4) 0 (mod 5), p(7n + 5) 0 (mod 7), p(11n + 6) 0 (mod 11). Motivated by Ramanujan s work, the congruence properties of partitions with certain restrictions have received a great deal of attention. In this talk, we will discuss congruence properties for Ramanujan s φ function. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

6 Ramanujan s φ function On page 3 of his lost notebook, Ramanujan 1 defined the function φ(q) as φ(q) = a(n)q n := which is not a Ramanujan s mock theta function. ( q;q) 2n q n+1 (q;q 2 ) 2, n+1 1 S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

7 Ramanujan s φ function On page 3 of his lost notebook, Ramanujan 1 defined the function φ(q) as φ(q) = a(n)q n := ( q;q) 2n q n+1 (q;q 2 ) 2, n+1 which is not a Ramanujan s mock theta function. Here and throughout this paper, we assume q < 1 for convergence and let n 1 (a;q) 0 = 1, (a;q) n = (1 aq k ), (a;q) = (1 aq k ), k=0 k=0 (a 1,a 2,...,a k ;q) = (a 1 ;q) (a 2 ;q) (a k ;q). 1 S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

8 Identities involving φ(q) Ramanujan also discovered the following identity involving φ(q 3 ) and a sixth order mock theta function ρ(q): ρ(q) = 2q 1 φ(q 3 ) + (q2 ;q 2 ) 2 ( q 3 ;q 3 ) (q;q 2 ) 2 (q 3 ;q 3 ), where ρ(q) = ( q;q) n q n(n+1)/2 (q;q 2 ) n+1. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

9 Identities involving φ(q) Choi 2 established two analogous identities involving φ(q) and the following two sixth order mock theta functions µ(q) and λ(q): µ(q) = λ(q) = ( 1) n q (n+1)2 (q;q 2 ) n ( q;q) 2n+1, ( 1) n q n (q;q 2 ) n ( q;q) n. 2 Y.-S. Choi, Identities for Ramanujan s sixth-order mock theta functions, Quart. J. Math. 53 (2002) K. Hikami, Transformation formula of the second order mock theta function, Lett. Math. Phys. 75 (2006) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

10 Identities involving φ(q) Choi 2 established two analogous identities involving φ(q) and the following two sixth order mock theta functions µ(q) and λ(q): µ(q) = λ(q) = ( 1) n q (n+1)2 (q;q 2 ) n ( q;q) 2n+1, ( 1) n q n (q;q 2 ) n ( q;q) n. Hikami 3 studied the transformation formula for φ(q). 2 Y.-S. Choi, Identities for Ramanujan s sixth-order mock theta functions, Quart. J. Math. 53 (2002) K. Hikami, Transformation formula of the second order mock theta function, Lett. Math. Phys. 75 (2006) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

11 Chan s congruences Chan 4 established a number of congruences for the coefficients a(n) of φ(q). 4 S.H. Chan, Congruences for Ramanujan s φ function, Acta Arith. 153 (2012) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

12 Chan s congruences Chan 4 established a number of congruences for the coefficients a(n) of φ(q). He proved that Theorem Theorem 1. For n 0, a(9n + 4) 0 (mod 2), a(18n + 10) a(25n + 14) a(25n + 24) 0 (mod 4), a(3n + 2) a(18n + 7) a(18n + 13) 0 (mod 3), a(10n + 9) 0 (mod 5), a(7n + 3) a(7n + 4) a(7n + 6) 0 (mod 7), a(6n + 5) 0 (mod 27). 4 S.H. Chan, Congruences for Ramanujan s φ function, Acta Arith. 153 (2012) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

13 Chan s conjecture Moreover, Chan conjectured that Conjecture Conjecture 1. For n 0, a(50n + j) 0 (mod 25), j {19,39,49}. 5 N. D. Baruah and N.M. Begum, Proofs of Some Conjectures of Chan on Appell-Lerch Sums, arxiv: Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

14 Chan s conjecture Moreover, Chan conjectured that Conjecture Conjecture 1. For n 0, a(50n + j) 0 (mod 25), j {19,39,49}. Baruah and Begum 5 confirmed Chan s conjecture and proved that Theorem Theorem 2. For n 0, a(1250n + 250r + 219) 0 (mod 125), r {1,3,4}. 5 N. D. Baruah and N.M. Begum, Proofs of Some Conjectures of Chan on Appell-Lerch Sums, arxiv: Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

15 The aim of this paper The aim of this paper is to prove new congruences modulo powers of 2, 3, 5 and 7 for Ramaujan s φ(q) function by utilizing some identities due to Newman 6,7 and theta function identities. 6 M. Newman, Modular forms whose coefficients possess multiplicative properties, Ann. Math. 70 (1959) M. Newman, Modular forms whose coefficients possess multiplicative properties (II), Ann. Math. 75 (1962) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

16 The aim of this paper The aim of this paper is to prove new congruences modulo powers of 2, 3, 5 and 7 for Ramaujan s φ(q) function by utilizing some identities due to Newman 6,7 and theta function identities. Method: find A, B and M such that a(an + B)q n eta quotients (mod M). Then, we utilize Ramanujan s theta function identities and Newman s identities to discover new congruences for a(n). 6 M. Newman, Modular forms whose coefficients possess multiplicative properties, Ann. Math. 70 (1959) M. Newman, Modular forms whose coefficients possess multiplicative properties (II), Ann. Math. 75 (1962) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

17 Congruences modulo powers of 2 Theorem Theorem 3. For n 0, ( 1) k k(3k + 1) (mod 4), if n = a(18n + 1) 2 0 (mod 4), otherwise, and for some k, (1) { a(n) + 1 (mod 2), if n 1 = 3k(k + 1) for some k, a(9n 2) a(n) (mod 2), otherwise. (2) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

18 Congruences modulo powers of 2 Corollary Corollary 1. For n,k 0, ( a 2 9 k+1 n + 5 ) 32k (mod 2). (3) 4 Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

19 Congruences modulo powers of 2 Corollary Corollary 1. For n,k 0, ( a 2 9 k+1 n + 5 ) 32k (mod 2). (3) 4 Example. Setting k = 1 in (3), we see that a(162n + 34) 0 (mod 2). Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

20 Congruences modulo powers of 2 Corollary Corollary 1. For n,k 0, ( a 2 9 k+1 n + 5 ) 32k (mod 2). (3) 4 Example. Setting k = 1 in (3), we see that Theorem a(162n + 34) 0 (mod 2). Theorem 4. a(9n + 4) is almost always divisible by 4 and a(18n + 10) is almost always divisible by 8. Remark. Chan proved that for n 0, a(9n + 4) 0 (mod 2) and a(18n + 10) 0 (mod 4). Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

21 Proofs of congruences modulo powers of 2 Chan proved the following identity: a(n)q n = (q18 ;q 18 ) (q 9 ;q 9 ) 2 1 (q 18 ;q 18 ) 2 (q 9 ;q 9 ) 2 n= n= ( 1) n q 9n2 +18n+9 1 q 18n+15 + q (q2 ;q 2 ) (q;q) 2 (q6 ;q 6 ) 4 (q 3 ;q 3 ) 2 ( 1) n q 9n2 +18n+7 1 q 18n+9 + q 2(q2 ;q 2 ) (q;q) 2 (q6 ;q 6 ) 3 (q18 ;q 18 ) 3 (q 3 ;q 3 ) (q 9 ;q 9 ) 3. (4) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

22 Proofs of congruences modulo powers of 2 Hirschhorn and Sellers 8 and Fortin, Jacob and Mathieu 9 proved that (q 2 ;q 2 ) (q;q) 2 = (q6 ;q 6 ) 4 (q 9 ;q 9 ) 6 (q 3 ;q 3 ) 8 (q 18 ;q 18 ) 3 + 2q (q6 ;q 6 ) 3 (q 9 ;q 9 ) 3 (q 3 ;q 3 ) 7 + 4q 2(q6 ;q 6 ) 2 (q 18 ;q 18 ) 3 (q 3 ;q 3 ) 6. (5) 8 M.D. Hirschhorn and J.A. Sellers, Arithmetic relations for overpartitions, J. Comb. Math. Comb. Comp. 53 (2005) J.-F. Fortin, P. Jacob and P. Mathieu, Jagged partitions, Ramanujan J. 10 (2005) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

23 Proofs of congruences modulo powers of 2 Hirschhorn and Sellers 8 and Fortin, Jacob and Mathieu 9 proved that (q 2 ;q 2 ) (q;q) 2 = (q6 ;q 6 ) 4 (q 9 ;q 9 ) 6 (q 3 ;q 3 ) 8 (q 18 ;q 18 ) 3 + 2q (q6 ;q 6 ) 3 (q 9 ;q 9 ) 3 (q 3 ;q 3 ) 7 + 4q 2(q6 ;q 6 ) 2 (q 18 ;q 18 ) 3 (q 3 ;q 3 ) 6. (5) If we substitute (5) into (4), extract those terms in which the power of q is congruent to 1 modulo 3, then divide by q and replace q 3 by q, we deduce that 8 M.D. Hirschhorn and J.A. Sellers, Arithmetic relations for overpartitions, J. Comb. Math. Comb. Comp. 53 (2005) J.-F. Fortin, P. Jacob and P. Mathieu, Jagged partitions, Ramanujan J. 10 (2005) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

24 Proofs of congruences modulo powers of 2 a(3n + 1)q n = (q2 ;q 2 ) 8 (q 3 ;q 3 ) 6 (q;q) 10 (q6 ;q 6 ) 3 1 (q 6 ;q 6 ) 2 (q 3 ;q 3 ) 2 n= + 4q (q2 ;q 2 ) 5 (q 6 ;q 6 ) 6 (q;q) 7 (q3 ;q 3 ) 3 ( 1) n q 3n2 +6n+2 1 q 6n+3. (6) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

25 Proofs of congruences modulo powers of 2 a(3n + 1)q n = (q2 ;q 2 ) 8 (q 3 ;q 3 ) 6 (q;q) 10 (q6 ;q 6 ) 3 1 (q 6 ;q 6 ) 2 (q 3 ;q 3 ) 2 n= + 4q (q2 ;q 2 ) 5 (q 6 ;q 6 ) 6 (q;q) 7 (q3 ;q 3 ) 3 ( 1) n q 3n2 +6n+2 1 q 6n+3. (6) By the binomial theorem, for any prime p and any positive integer k (q;q) pk (qp ;q p ) pk 1 (mod pk ). (7) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

26 Proofs of congruences modulo powers of 2 Employing the above congruence and some theta function identities, a(9n + 1)q n (q;q)4 (q3 ;q 3 ) 4 (q 2 ;q 2 ) (q 6 ;q 6 ) 2 (mod 8), (8) a(9n + 4)q n 2 (q;q)5 (q 3 ;q 3 ) (q 6 ;q 6 ) (q 2 ;q 2 ) 2 (mod 4), (9) and a(9n + 7)q n (q;q)6 (q6 ;q 6 ) 4 (q 2 ;q 2 ) 3 (q3 ;q 3 ) 2 1 (q 2 ;q 2 ) 2 (q;q) 2 n= ( 1) n q n2 +2n 1 q 2n+1 (mod 2). (10) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

27 Proofs of congruences modulo powers of 2 Thanks to (7) and (8), a(18n + 1)q n (q;q) (mod 4). (11) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

28 Proofs of congruences modulo powers of 2 Thanks to (7) and (8), a(18n + 1)q n (q;q) (mod 4). (11) The following identity is commonly known as Euler s pentagonal number theorem and is worth highlighting here: (q;q) = k= Congruence (1) follows from (11) and (12). ( 1) k q k(3k+1) 2. (12) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

29 Proofs of congruences modulo powers of 2 Thanks to (7) and (8), a(18n + 1)q n (q;q) (mod 4). (11) The following identity is commonly known as Euler s pentagonal number theorem and is worth highlighting here: (q;q) = k= Congruence (1) follows from (11) and (12). It follows from (10) and the definition of a(n) that a(9n + 7)q n+1 which yields (2). k=0 ( 1) k q k(3k+1) 2. (12) q 3k(k+1)+1 + a(n)q n (mod 2), (13) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

30 Proofs of congruences modulo powers of 2 Replacing n by 2n in (2) yields a(18n 2) a(2n) (mod 2) (14) By (14) and mathematical induction, we find that for n, k 0, ( ) a 2 9 k n 9k 1 a(2n) (mod 2) (15) 4 Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

31 Proofs of congruences modulo powers of 2 Replacing n by 2n in (2) yields a(18n 2) a(2n) (mod 2) (14) By (14) and mathematical induction, we find that for n, k 0, Chan proved that for n 0, ( ) a 2 9 k n 9k 1 a(2n) (mod 2) (15) 4 a(18n + 4) 0 (mod 2). (16) Replacing n by 9n + 2 in (15) and employing (16), we arrive at (3). Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

32 Proofs of congruences modulo powers of 2 Lemma Lemma 1. We have a(9n + 4)q n 2(q;q) (q 6 ;q 6 ) 2 (q 3 ;q 3 ) (mod 4), (17) a(18n + 10)q n 4(q 4 ;q 4 ) (q 6 ;q 6 ) 2 (q 3 ;q 3 ) (mod 8). (18) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

33 Proofs of congruences modulo powers of 2 Lemma Lemma 1. We have a(9n + 4)q n 2(q;q) (q 6 ;q 6 ) 2 (q 3 ;q 3 ) (mod 4), (17) a(18n + 10)q n 4(q 4 ;q 4 ) (q 6 ;q 6 ) 2 (q 3 ;q 3 ) (mod 8). (18) It follows that a(9n + 4)q 24n+10 2 a(18n + 10)q 24n+13 4 k= m=0 k= m=0 q (6k+1)2 +3(2m+1) 2 (mod 4), q (12k+2)2 +3(2m+1) 2 (mod 8). Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

34 Proofs of congruences modulo powers of 2 Therefore, and a(9n + 4) 2r a 2 +3b 2, b>0(24n + 10) (mod 4) a(18n + 10) 4r a 2 +3b 2, b>0(24n + 13) (mod 8), where r a 2 +3b2(n) is the number of representations of n as the sum of a square and three times a square. Now, r a 2 +3b2(n) can be given in terms of the divisors of n (see Berndt s book 10 ), but the fact is, that, as with all quadratic forms in two variables, the number of representations is almost always 0. Theorem 4 follows. 10 B.C. Berndt, Number Theory in the Spirit of Ramanujan, American Mathematical Society, Providence, RI Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

35 New congruences modulo powers of 2 Theorem Theorem 5. For n 0, and a(49n 12) a(49n + 9) a(49n + 23) a(49n + 30) 0 (mod 2) { a(n) + 1 (mod 2), if n 2 = 7k(k + 1) for some k, a(n) (mod 2), otherwise. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

36 New congruences modulo powers of 2 Corollary Corollary 2. For n, k 0, ( a 2 49 k+1 n + (8s + 3) ) 49k (mod 2), 4 where s {4, 11, 23}. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

37 New congruences modulo powers of 2 Corollary Corollary 2. For n, k 0, ( a 2 49 k+1 n + (8s + 3) ) 49k (mod 2), 4 where s {4, 11, 23}. Example. Taking k = 1 in the above congruence, we see that for n 0, a(4802n + 429) a(4802n ) a(4802n ) 0 (mod 2). Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

38 New congruences modulo powers of 3 Theorem Theorem 6. For n 0, a(54n + 25) a(54n + 43) a(162n + 61) a(162n + 115) 0 (mod 9). and 3( 1) k k(3k + 1) (mod 9), if n = a(162n + 7) 2 0 (mod 9), otherwise, for some ineteger k, Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

39 New congruences modulo powers of 3 Theorem Theorem 6. For n 0, a(54n + 25) a(54n + 43) a(162n + 61) a(162n + 115) 0 (mod 9). and 3( 1) k k(3k + 1) (mod 9), if n = a(162n + 7) 2 0 (mod 9), otherwise, for some ineteger k, Theorems 5, 6 and Corollary 2 can be proved similarly, so we omit the details. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

40 New congruences modulo powers of 3 In order to state congruences for a(n), we introduce ( ) the Legendre symbol. x Let p 3 be a prime. The Legendre symbol is defined by p L ( ) x 1, if x is a quadratic residue modulo p and p x, := 0, if p x, p L 1, if x is a nonquadratic residue modulo p. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

41 New congruences modulo powers of 3 In order to state congruences for a(n), we introduce ( ) the Legendre symbol. x Let p 3 be a prime. The Legendre symbol is defined by p L ( ) x 1, if x is a quadratic residue modulo p and p x, := 0, if p x, p L 1, if x is a nonquadratic residue modulo p. Let b 1 (n) be defined by b 1 (n)q n := (q;q) 2 (q3 ;q 3 ) and let p 5 be a prime. Define ω 1 (p) := b 1 ( 5(p 2 1) 24 ) ( p p Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42 ) L.

42 New congruences modulo 3 Theorem Theorem 7. (1) For n, k 0, if p n, then ( ) a 18p α1(p)(k+1) 1 n + 15pα1(p)(k+1) (mod 3), 4 where 4, if ω 1 (p) 0 (mod 3), α 1 (p) : = 6, if ω 1 (p) 0 (mod 3) and p 1 (mod 3), 8, if ω 1 (p) 0 (mod 3) and p 2 (mod 3). Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

43 New congruences modulo 3 Theorem ( ) 6n 1+ (2) For n, k 0, if ω 1 (p) p2 1 4 p 1 (mod 3), then L ( ) a 18p α1(p)k+2 n + 15pα1(p)k (mod 3). 4 Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

44 New congruences modulo 3 Theorem ( ) 6n 1+ (2) For n, k 0, if ω 1 (p) p2 1 4 p 1 (mod 3), then L ( ) a 18p α1(p)k+2 n + 15pα1(p)k (mod 3). 4 Example. Setting p = 7, k = 0 in Theorem 7 and using the fact that α 1 (7) = 6, we find that for n 0, and a( (7n + j) ) 0 (mod 3), 1 j 5 with j 0 a(6174n + 184) a(6174n ) a(6174n ) 0 (mod 3). Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

45 New congruences modulo 3 The proof of Theorem 7 mainly relies on Newman s identity and the following congruence: a(18n + 4)q n 2(q;q) 2 (q3 ;q 3 ) (mod 3). Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

46 New congruences modulo powers of 5 Let b 2 (n) be defined by b 2 (n)q n := (q;q) 5 (q 2 ;q 2 ) 8 and let p 7 be a prime. Define ( 7(p 2 ) 1) ω 2 (p) := b 2 + p 8 ( ) 1 3(p 2 1) 4. p L Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

47 New congruences modulo powers of 5 Let b 2 (n) be defined by b 2 (n)q n := (q;q) 5 (q 2 ;q 2 ) 8 and let p 7 be a prime. Define ( 7(p 2 ) 1) ω 2 (p) := b 2 + p 8 ( ) 1 3(p 2 1) 4. p L Assume that ω 2 (p) i (mod 5) and p 3 j (mod 5) with 0 i 4 and 1 j 4. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

48 New congruences modulo powers of 5 Define S 1 :={(0,1),(0,2),(0,3),(0, 4)}, S 2 :={(1,4),(2,1),(3,1),(4, 4)}, S 3 :={(1,2),(2,3),(3,3),(4, 2)}, S 4 :={(1,1),(2,4),(3,4),(4, 1)}, S 5 :={(1,3),(2,2),(3,2),(4, 3)}. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

49 New congruences modulo powers of 5 Define S 1 :={(0,1),(0,2),(0,3),(0, 4)}, S 2 :={(1,4),(2,1),(3,1),(4, 4)}, S 3 :={(1,2),(2,3),(3,3),(4, 2)}, S 4 :={(1,1),(2,4),(3,4),(4, 1)}, S 5 :={(1,3),(2,2),(3,2),(4, 3)}. Furthermore, define α 2 (p) := u + 1 if (i,j) S u for 1 u 5. (19) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

50 New congruences modulo powers of 5 Theorem Theorem 8. (1) For n, k 0, if p n, then ( ) a 10p 2α2(p)(k+1) 1 n + 35p2α2(p)(k+1) (mod 25), 4 and a ( 250p α 2(p)(k+1) 1 n + 875pα 2(p)(k+1) ) 0 (mod 125). Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

51 New congruences modulo powers of 5 Theorem ( ) 2n+1 (2) For n, k 0, if with ω 2 (p) 3(p2 1) 4 p p (mod 5), then L ( ) a 10p 2α2(p)k+2 n + 35p2α2(p)k (mod 25) 4 and a ( 250p 2α 2(p)k+2 n + 875p2α 2(p)k ) 0 (mod 125). Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

52 New congruences modulo powers of 5 Theorem ( ) 2n+1 (2) For n, k 0, if with ω 2 (p) 3(p2 1) 4 p p (mod 5), then L ( ) a 10p 2α2(p)k+2 n + 35p2α2(p)k (mod 25) 4 and a ( 250p 2α 2(p)k+2 n + 875p2α 2(p)k ) 0 (mod 125). Example. Taking k = 0 and p = 7 in Theorem 8 yields a( n ) 0 (mod 25), (7 n), a( n ) 0 (mod 125), (7 n). Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

53 New congruences modulo powers of 5 The proof of Theorem 8 mainly relies on Newman s identity and the following congruences: and a(10n + 9)q n 5(q;q) 5 (q 2 ;q 2 ) 8 (mod 25) a(250n + 219)q n 50(q;q) 5 (q2 ;q 2 ) 8 (mod 125). The above two congruences were proved by Baruah and Begum N. D. Baruah and N.M. Begum, Proofs of Some Conjectures of Chan on Appell-Lerch Sums, arxiv: Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

54 New congruences modulo 7 Define and b 3 (n)q n := (q2 ;q 2 ) 8 (q;q) ( 5(p 2 ) ( ) 1) 5 ω 3 (p) := b 3 + p 2, 8 p L where p 5 is a prime and p 7. Assume that ω 3 (p) i (mod 7) and p 5 j (mod 7) with 0 i 6 and 1 j 6. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

55 New congruences modulo 7 Define W 1 :={(0,1),(0,2),(0,3),(0,4), (0, 5),(0,6)}, W 2 :={(1,6),(2,3),(3,5),(4,5), (5, 3),(6,6)}, W 3 :={(1,3),(2,5),(3,6),(4,6), (5, 5),(6,3)}, W 4 :={(1,2),(2,1),(3,4),(4,4), (5, 1),(6,2)}, W 5 :={(1,5),(2,6),(3,3),(4,3), (5, 6),(6,5)}, W 6 :={(1,1),(1,4),(2,2),(2,4), (3, 1),(3,2),(4, 1), (4,2),(5,2), (5, 4),(6,1), and { k + 1, if (i,j) Wk where k {1, 2, 3}, α 3 (p) := k + 2, if (i,j) W k where k {4, 5, 6}. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

56 New congruences modulo 7 Theorem Theorem 9. Let p 5 be a prime with p 7. For k, n 0, if p n, then ( ) a 14p 2α3(p)(k+1) 1 n + 35p2α3(p)(k+1) (mod 7). (20) 4 ( ) 2n 1+ For k, n 0, if ω 3 (p) p2 1 4 p p 2 (mod 7), then L a ( 14p 2α 3(p)k+2 n + 35p2α 3(p)k ) 0 (mod 7). (21) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

57 New congruences modulo 7 Example. It is easy to check that ω 3 (5) = 54 5 (mod 7) and (mod 7). Therefore, α 3 (5) = 8. Setting p = 5 and k = 0 in Theorem 9, we have a( n ) 0 (mod 7) (5 n). Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

58 Proof of Theorem 9 We only prove Theorem 9. Since Theorems 7 and 8 can be proved similarly, we omit the details. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

59 Proof of Theorem 9 We only prove Theorem 9. Since Theorems 7 and 8 can be proved similarly, we omit the details. Lemma Lemma 2. We have a(14n + 9)q n (q2 ;q 2 ) 8 (q;q) (mod 7). This lemma can be proved by using some theta function identities. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

60 Proof of Theorem 9 We only prove Theorem 9. Since Theorems 7 and 8 can be proved similarly, we omit the details. Lemma Lemma 2. We have a(14n + 9)q n (q2 ;q 2 ) 8 (q;q) (mod 7). This lemma can be proved by using some theta function identities. From Lemma 2, a(14n + 9) b 3 (n) (mod 7). (22) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

61 Proof of Theorem 9 Lemma Lemma 3. Let p 5 be a prime with p 7. For n, k 0, ( ) ( ) b 3 p 2k n + 7(p2k 1) U p (k)b 3 p 2 n + 7(p2 1) 8 8 where U p (k) = ω 3 (p)u p (k 1) p 5 U p (k 2), V p (k) = ω 3 (p)v p (k 1) p 5 V p (k 2), with U p (0) = V p (1) = 0 and U p (1) = V p (0) = 1. + V p (k)b 3 (n) (mod 7), (23) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

62 Proof of Theorem 9 The above lemma can be proved by induction and the following identity due to Newman: ( ) ( ) b 3 p 2 n + 7(p2 1) n 7(p 2 1) = χ(n)b 3 (n) p 5 8 b 3 8 p 2, where χ(n) is a function in n and p. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

63 Proof of Theorem 9 The above lemma can be proved by induction and the following identity due to Newman: ( ) ( ) b 3 p 2 n + 7(p2 1) n 7(p 2 1) = χ(n)b 3 (n) p 5 8 b 3 8 p 2, where χ(n) is a function in n and p. By Lemma 3 and the above identity, ( ) ( ) b 3 p 2k+1 n + 7(p2k+2 1) (ω 3 (p)u p (k) + V p (k))b 3 pn + 7(p2 1) 8 8 p 5 U p (k)b 3 (n/p) (mod 7). (24) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

64 Proof of Theorem 9 Lemma Lemma 4. For k 0 ω 3 (p)u p (k) + V p (k) U p (k + 1) (mod 7). Lemma 4 can be proved by induction. We omit the details. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

65 Proof of Theorem 9 Lemma Lemma 4. For k 0 ω 3 (p)u p (k) + V p (k) U p (k + 1) (mod 7). Lemma 4 can be proved by induction. We omit the details. In view of (24) and Lemma 4, ( ) ( ) b 3 p 2k+1 n + 7(p2k+2 1) U p (k + 1)b 3 pn + 7(p2 1) 8 8 p 5 U p (k)b 3 (n/p) (mod 7). (25) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

66 Proof of Theorem 9 Lemma Lemma 5. For k 0, U p (α 3 (p)k) 0 (mod 7). Lemma 5 can be proved by induction. We omit the details. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

67 Proof of Theorem 9 Lemma Lemma 5. For k 0, U p (α 3 (p)k) 0 (mod 7). Lemma 5 can be proved by induction. We omit the details. Replacing k by α 3 (p)(k + 1) 1 in (25) and utilizing Lemma 5, we see that for n, k 0, ( ) b 3 p 2α3(p)(k+1) 1 n + 7(p2α3(p)(k+1) 1) 8 p 5 U p (α 3 (p)(k + 1) 1)b 3 (n/p) (mod 7). (26) Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

68 Proof of Theorem 9 Lemma Lemma 5. For k 0, U p (α 3 (p)k) 0 (mod 7). Lemma 5 can be proved by induction. We omit the details. Replacing k by α 3 (p)(k + 1) 1 in (25) and utilizing Lemma 5, we see that for n, k 0, ( ) b 3 p 2α3(p)(k+1) 1 n + 7(p2α3(p)(k+1) 1) 8 If p n, then n p p 5 U p (α 3 (p)(k + 1) 1)b 3 (n/p) (mod 7). (26) is not an integer and b 3 (n/p) = 0. (27) Combining (22), (26) and (27), we arrive at the first part of Theorem 9. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

69 Proof of Theorem 9 ( ) 2n 1+ If ω 3 (p) p2 1 4 p p 2 (mod 7), then L ( ) n 7(p2 1) ( ) 8 b 3 = 0, b 3 p 2 n + 7(p2 1) 0 (mod 7). (28) p 8 Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

70 Proof of Theorem 9 ( ) 2n 1+ If ω 3 (p) p2 1 4 p p 2 (mod 7), then L ( ) n 7(p2 1) ( ) 8 b 3 = 0, b 3 p 2 n + 7(p2 1) 0 (mod 7). (28) p 8 Replacing k by α 3 (p)k in (23) and utilizing Lemma 5 yields ( ) b 3 p 2α3(p)k n + 7(p2α3(p)k 1) V p (α 3 (p)k)b 3 (n) (mod 7). (29) 8 Replacing n by p 2 n + 7(p2 1) 8 in (29), then employing (22) and (28), we arrive at the second part of Theorem 9. This completes the proof. Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42

71 Thank you for your attention! Ernest X.W. Xia (Jiangsu University) Arithmetic Properties for Ramanujan s φ function August 6, / 42