A Family of Supercongruences Involving Multiple Harmonic Sums
aa r X i v : . [ m a t h . N T ] J a n A FAMILY OF SUPERCONGRUENCES INVOLVING MULTIPLEHARMONIC SUMS
MEGAN MCCOY ∗ , KEVIN THIELEN ∗ , LIUQUAN WANG † , AND JIANQIANG ZHAO ⋆ Abstract.
In recent years, the congruence X i + j + k = pi,j,k> ijk ≡ − B p − (mod p ) , first discovered by the last author have been generalized by either increasing the numberof indices and considering the corresponding supercongruences, or by considering the al-ternating version of multiple harmonic sums. In this paper, we prove a family of similarsupercongruences modulo prime powers p r with the indexes summing up to mp r where m iscoprime to p , where all the indexes are also coprime to p . Introduction
Multiple harmonic sums are multiple variable generalization of harmonic numbers. Let N be the set of natural numbers. For s = ( s , . . . , s d ) ∈ N d and any N ∈ N , we define themultiple harmonic sums (MHS) by H N ( s ) := X N ≥ k > ··· >k d > d Y i =1 k s i i . Since mid 1980s these sums have appeared in a few diverse areas of mathematics as well astheoretical physics such as multiple zeta values [4, 5, 7], Feynman integrals [1, 3], quantumelectrodynamics and quantum chromodynamics [2, 10].In [17] the last author started to investigate congruence properties of MHSs, which were alsoconsidered by Hoffman [5] independently. As a byproduct, the following intriguing congruencewas noticed: for all primes p ≥ X i + j + k = pi,j,k> ijk ≡ − B p − (mod p ) , (1)where B k are Bernoulli numbers defined by the generating series te t − ∞ X k =0 B k t k k ! . This was proved by the last author using MHSs in [16], and by Ji using some combinatorialidentities in [6]. Later on, a few generalizations and analogs were obtained by either increasing
Mathematics Subject Classification.
Key words and phrases.
Multiple harmonic sums, Bernoulli numbers, Supercongruences. ∗ , KEVIN THIELEN ∗ , LIUQUAN WANG † , AND JIANQIANG ZHAO ⋆ the number of indices and considering the corresponding supercongruences (see [11, 13, 15,21]), or considering the alternating version of MHSs (see [9, 12]).Let P n be the set of positive integers not divisible by n . To generalize the congruence in(1), we wonder if for every odd integer d ≥ q d such that X l + l + ··· + l d = p r l ,...,l d ∈P p l l . . . l d ≡ q d · p r − B p − d (mod p r ) (2)for any prime p > d and integer r ≥
2. In [11, 13] it is shown that q = − q = − / d is even, the congruence pattern is quite different, see[14, 19]. In this paper, we shall prove the following main result when d = 7. Theorem 1.1.
Let r and m be positive integers and p > be a prime such that p ∤ m . (i) If r = 1 , then X l + l + ··· + l = mpl ,...,l ∈P p l l . . . l ≡ − (504 m + 210 m + 6 m ) B p − (mod p ) . (ii) If r ≥ , then X l + l + ··· + l = mp r l ,...,l ∈P p l l . . . l ≡ − · mp r − B p − (mod p r ) . To establish this result, for all positive integers n , m , r and primes p , following the notationin [13], we define S ( m ) n ( p r ) := X l + l + ··· + l n = mp r p r >l ,...,l n ∈P p l l . . . l n . Notice that the sum in the theorem is not exactly the same type as that appearing in S ( m ) n since the condition p r > l i for all i is not present. The main idea of our proof is to show thespecial case when m = 1 first. In order to do this we will first prove the relation S (1) n ( p r +1 ) ≡ pS (1) n ( p r ) (mod p r +1 ) , ∀ r ≥ , (3)and then use induction. Notice that when r = 1 the above congruence usually does not holdanymore. So we will compute the congruence of S (1) n ( p ) and S (1) n ( p ) separately by relatingthem to the following quantities: R ( m ) n ( p ) := X l + l + ··· + l n = mpl ,...,l n ∈P p l l . . . l n . To save space, throughout the paper when the prime p is fixed we often use the shorthand H ( s ) = H p − ( s ). Moreover, we shall also need the modified sum H ( p ) N ( s ) := X N ≥ k > ··· >k d > k ,...,k d ∈P p d Y i =1 k s i i . FAMILY OF SUPERCONGRUENCES INVOLVING MULTIPLE HARMONIC SUMS 3 Preliminary lemmas
Let C ( m ) a,p ( n ) denote the number of solutions ( x , . . . , x n ) of the equation x + · · · + x n = mp − a, ≤ x i < p ∀ i = 1 , . . . , n. For all b ≥ β n ( a, b ) := (cid:18) bp − a + n − n − (cid:19) and γ n ( a ) := ( − a − a (cid:0) n − a (cid:1) . It is not hard to see that β n ( a, b ) ≡ b ( − a − ( n − a − a − n − p ≡ b ( − a − a (cid:0) n − a (cid:1) p ≡ bγ n ( a ) p (mod p ) . (4) Lemma 2.1.
For all m, n, a ∈ N and primes p , we have C ( m ) a,p ( n ) ≡ ( − m − (cid:18) n − m − (cid:19) γ n ( a ) p ≡ ( − m − (cid:18) n − m − (cid:19) C (1) a,p ( n ) (mod p ) . Proof.
The coefficient of x mp − a in the expansion of (cid:0) x + · · · + x p − (cid:1) n = ( x p − n ( x − − n is C ( m ) a,p ( n ) = m X i =0 (cid:18) ni (cid:19)(cid:18) − nmp − ip − a (cid:19) ( − mp − a = m X i =0 (cid:18) ni (cid:19) ( − ip (cid:18) n + mp − ip − a − n − (cid:19) = m X i =0 ( − i (cid:18) ni (cid:19)(cid:18) n + mp − ip − a − n − (cid:19) ≡ m X i =0 ( − i (cid:18) ni (cid:19) ( m − i ) γ n ( n − a ) p (mod p )by (4). Now we calculate the sum A ( m ) = m X i =0 ( − i (cid:18) ni (cid:19) ( m − i ) . It is easy to see that A ( m ) is the coefficient of x m in the expansion of(1 − x ) n · ∞ X i =0 ix i = (1 − x ) n · x (1 − x ) = x (1 − x ) n − = n − X m =1 ( − m (cid:18) n − m − (cid:19) x m , as desired. (cid:3) Corollary 2.2.
When n = 7 , we have C (2)1 ,p (7) − C (2)6 ,p (7) ≡ − (5 / p, C (3)1 ,p (7) − C (3)6 ,p (7) ≡ (10 / p (mod p ) ,C (3)2 ,p (7) − C (3)5 ,p (7) ≡ − (2 / p, C (2)2 ,p (7) − C (2)5 ,p (7) ≡ (1 / p (mod p ) ,C (3)3 ,p (7) − C (3)4 ,p (7) ≡ (1 / p, C (2)3 ,p (7) − C (2)4 ,p (7) ≡ − (1 / p (mod p ) . Part (ii) of the following lemma generalizes [13, Lemma 1(ii)].
MEGAN MCCOY ∗ , KEVIN THIELEN ∗ , LIUQUAN WANG † , AND JIANQIANG ZHAO ⋆ Lemma 2.3.
Let ≤ k ≤ n − and p > n a prime. For all r ≥ , we have (i) S ( k ) n ( p r ) ≡ ( − n S ( n − k ) n ( p r ) (mod p r ) . (ii) S ( m ) n ( p r +1 ) ≡ n − X a =1 C ( m ) a,p ( n ) S ( a ) n ( p r ) (mod p r +1 ) . Proof. (i) can be found in [13]. We now prove (ii). For any n -tuples ( l , · · · , l n ) of integerssatisfying l + · · · + l n = mp r +1 , p r +1 > l i ∈ P p , 1 ≤ i ≤ n , we rewrite them as l i = x i p r + y i , ≤ x i < p, ≤ y i < p r , y i ∈ P p , ≤ i ≤ n. Since (cid:16) n X i =1 x i (cid:17) p r + n X i =1 y i = mp r +1 and n < p , we know there exists 1 ≤ a < n such that (cid:26) x + · · · + x n = mp − a, ≤ x i < p,y + · · · + y n = ap r . For 1 ≤ a < n , the equation x + · · · + x n = mp − a has C ( m ) a,p ( n ) integer solutions with0 ≤ x i < p . Hence by Lemma 2.1 S ( m ) n ( p r +1 ) = X l + ··· + l n = mp r +1 l , ··· ,l n ∈P p l l · · · l n = n − X a =1 X x + ··· + x n = mp − a ≤ x i
We first consider some un-ordered sums. Lemmas 3.1 and 3.3 were proved by Zhou andCai [21].
FAMILY OF SUPERCONGRUENCES INVOLVING MULTIPLE HARMONIC SUMS 5
Lemma 3.1.
Let p be a prime and α , . . . , α n be positive integers, r = α + · · · + α n ≤ p − .Define the un-ordered sum U b ( α , . . . , α n ) = X Corollary 3.2. Let p be a prime and α be positive integer. Then H ( { α } n ) ≡ ( − n α ( nα + 1)2( nα + 2) B p − nα − · p (mod p ) , if nα is odd; ( − n − αnα + 1 B p − nα − · p (mod p ) , if nα is even. Lemma 3.3. Let n > be positive integer and let p > n + 1 be a prime. Then R (1) n ( p ) = X l + ··· + l n = pl ,...,l n > l · · · l n ≡ ( − ( n − B p − n (mod p ) , if n is odd; − n · n ! n + 1 B p − n − p (mod p ) , if n is even. The next result generalizes Lemma 3.1. Lemma 3.4. Let p be a prime and α , . . . , α n be positive integers, r = α + · · · + α n ≤ p − .Then U b ( α , . . . , α n ) ≡ ( − n ( n − b r ( r + 1)2( r + 2) B p − r − · p (mod p ) , if r is odd; ( − n − ( n − brr + 1 B p − r − · p (mod p ) , if r is even.Proof. For all k ≥ 1, we have X kp 12 + k (cid:19) B p − α − p (mod p ) , if α is odd; αα + 1 B p − α − p (mod p ) , if α is even. MEGAN MCCOY ∗ , KEVIN THIELEN ∗ , LIUQUAN WANG † , AND JIANQIANG ZHAO ⋆ Therefore for any positive integer b , we have X Let n be an odd positive integer and p be a prime. Then R (2) n ( p ) = X l + ··· + l n =2 pl ,...,l n ∈P p l . . . l n ≡ − n + 12 · ( n − B p − n (mod p ) . Proof. We have X l + ··· + l n =2 pl ,...,l n ∈P p l . . . l n = X l + ··· + l n =2 p l . . . l n − np X l + ··· + l n − = p l . . . l n − FAMILY OF SUPERCONGRUENCES INVOLVING MULTIPLE HARMONIC SUMS 7 = n !2 p X
Let n be an odd positive integer with n ≥ . Then for all prime p > n , wehave S (2) n ( p ) ≡ n − · ( n − B p − n (mod p ) . Proof. We observe that X l + ··· + l n =2 pl j ∈P p ∀ j l . . . l n ≡ X l + ··· + l n =2 pl ,...,l n
Let n ≥ be an odd positive integer. Then for all prime p ≥ max { n, } , wehave R (3) n ( p ) = X l + ··· + l n =3 pl ,...,l n ∈P p l . . . l n ≡ − n (cid:18) n + 23 (cid:19) · ( n − B p − n − n !6 X a + b + c = n − a,b,c ≥ B p − a − B p − b − B p − c − (2 a + 1)(2 b + 1)(2 c + 1) (mod p ) . MEGAN MCCOY ∗ , KEVIN THIELEN ∗ , LIUQUAN WANG † , AND JIANQIANG ZHAO ⋆ Proof. Let ν = n − u i = l + · · · + l i , 1 ≤ i ≤ ν . We have X l + ··· + l n =3 pl ,...,l n ∈P p l . . . l n = n !3 p X ≤ u < ··· j,u k ,u j +2 − u j +1 ,...,u ν − u n − ∈P p u j +1 · · · u ν = 1 p H ( { } j − ) X
Similarly, for 2 ≤ j ≤ n − X ≤ u < ···
4, we obtain X ≤ u < ···
We have D j = X ≤ v < ··· Let n ≥ be an odd positive integer. Then for all prime p ≥ max { n, } , wehave S (3) n ( p ) ≡ − n (cid:18) n (cid:19) · ( n − B p − n − n !6 X a + b + c = n − a,b,c ≥ B p − a − B p − b − B p − c − (2 a + 1)(2 b + 1)(2 c + 1) (mod p ) . Proof. We observe that X l + ··· + l n =3 pl j ∈P p ∀ j l . . . l n ≡ X l + ··· + l n =3 pl j
So we deduce S (3) n ( p ) ≡ X l + ··· + l n =3 pl j ∈P p ∀ j l . . . l n − (cid:18) n + 12 (cid:19) S (1) n ( p ) − nS (2) n ( p ) ≡ − n (cid:18) n (cid:19) · ( n − B p − n − n !6 X a + b + c = n − a,b,c ≥ B p − a − B p − b − B p − c − (2 a + 1)(2 b + 1)(2 c + 1) (mod p )by Lemma 3.7, since S (1) n ( p ) ≡ − ( n − B p − n (mod p ) by Lemma 3.3 and S (2) n ( p ) ≡ − n − ( n − B p − n (mod p ) by Corollary 3.6. (cid:3) Proof of the main theorem First, we prove a special case of Theorem 1.1. Proposition 4.1. For all r ≥ and prime p > we have S (1)7 ( p r +1 ) ≡ − B p − p r (mod p r +1 ) . Proof. By Lemma 2.3, for all r ≥ 1, we have S ( m ) n ( p r +1 ) ≡ n − X a =1 (cid:0) ( − m − (cid:18) n − m − (cid:19) γ n ( a ) p + O ( p ) (cid:1) S ( a ) n ( p r ) (mod p r +1 ) . Here the O ( p ) means a quantity which remains a p -adic integer after dividing by the p . Byinduction on r it is not hard to see that for all m = 1 , . . . , n − 1, we have S ( m ) n ( p r +1 ) ≡ p r ) , for all r ≥ . Thus for all m = 1 , . . . , n − 1, by Lemmas 2.1 and 2.3, we have S ( m ) n ( p r +1 ) ≡ n − X a =1 ( − m − (cid:18) n − m − (cid:19) γ n ( a ) pS ( a ) n ( p r ) (mod p r +1 ) ≡ ( − m − (cid:18) n − m − (cid:19) S (1) n ( p r +1 ) (mod p r +1 ) . Thus by Lemmas 2.1 and 2.3, for all r ≥ S (1) n ( p r +1 ) ≡ n − X m =1 C ma,p ( n ) S ( m ) n ( p r ) (mod p r +1 ) ≡ n − X m =1 ( − m − (cid:18) n − m − (cid:19) pγ n ( m ) S (1) n ( p r ) (mod p r +1 ) ≡ n − X m =1 ( n − m − m − p ( n − (cid:18) n − m − (cid:19) S (1) n ( p r ) (mod p r +1 ) ≡ pS (1) n ( p r ) (mod p r +1 ) , ∗ , KEVIN THIELEN ∗ , LIUQUAN WANG † , AND JIANQIANG ZHAO ⋆ which proves (3). Finally, by applying Lemma 2.3 when n = 7, we get S (1)7 ( p ) ≡ p S (1)7 ( p ) − p S (2)7 ( p ) + p S (3)7 ( p r ) (mod p ) ≡ (cid:18) − p − p − p (cid:19) B p − ≡ − B p − p (mod p )by Lemma 3.3, Corollary 3.6 and Corollary 3.8. (cid:3) We are now ready to prove Theorem 1.1.Let n = mp r , where p does not divide m . For any 7-tuples ( l , · · · , l ) of integers satisfying l + · · · + l = n , l i ∈ P p , 1 ≤ i ≤ 7, we rewrite them as l i = x i p r + y i , x i ≥ , ≤ y i < p r , y i ∈ P p , ≤ i ≤ . Since (cid:16) X i =1 x i (cid:17) p r + X i =1 y i = mp r , we know there exists 1 ≤ a ≤ (cid:26) x + · · · + x = m − a,y + · · · + y = ap r . For 1 ≤ a ≤ 6, the equation x + · · · + x = m − a has (cid:0) m +6 − a (cid:1) nonnegative integer solutions.Hence X l + ··· + l = mp r l , ··· ,l ∈P p l l · · · l = X a =1 X x + ··· + x = m − a X y + ··· + y = ap r y i ∈P p ,y i
2, then we have S (2)7 ( p r ) ≡ − S (1)7 ( p r ) (mod p r ) and S (3)7 ( p r ) ≡ S (1)7 ( p r )(mod p r ). Meanwhile, we have S ( a )7 ( p ) ≡ − S (7 − a )7 ( p ) (mod p r ) for 4 ≤ a ≤ 6. Hence from(12) we obtain X l + ··· + l = nl , ··· ,l ∈P p l l · · · l ≡ X a =0 ( − a (cid:18) a (cid:19)(cid:18) m + 5 − a (cid:19) S (1)7 ( p r ) ≡ mS (1)7 ( p r ) (mod p r ) . Since S (1)7 ( p r ) ≡ − p r − B p − (mod p r ) by Proposition 4.1, we complete the proof of (ii). FAMILY OF SUPERCONGRUENCES INVOLVING MULTIPLE HARMONIC SUMS 15 Concluding remarks Using similar ideas from [19] we find that it is unlikely to further generalize our main resultto congruence (2) for r ≥ 2, odd integer d ≥ 9, and q d ∈ Q depending only on d . By usingPSLQ algorithm we find that both the numerator and the denominator of q would have atleast 60 digits if the congruence (2) holds for every prime p ≥ 11. However, when r = 1 wehave obtained a few general congruences in Lemma 3.3, Lemma 3.5 and Corollary 3.6, whichcan be rephrased as follows. Let m = 1 , d be any odd integer greater than 2. Then forany prime p > d , we have S ( m ) d ( p ) ≡ c d,m · ( d − B p − d (mod p ) , (13)where c d, = − c d, = ( d − / 2, and R ( m ) d ( p ) ≡ c ′ d,m · ( d − B p − d (mod p ) , (14)where c ′ d, = − c ′ d, = − ( d + 1) / 2. Unfortunately, Lemma 3.7 and Corollary 3.8 implythat these do not generalize to m ≥ 3. Computation with PSLQ algorithm suggests thatif (13) and (14) hold for d = 9 , , , m = 3 , c d,m and c ′ d,m would have at least 60 digits. In fact, numerical evidencesuggests the following conjecture. Conjecture 5.1. For any prime p ≥ , we have R ( m )8 ( p ) ≡ m ( m + 16)( m − B p − B p − (mod p ) ,R ( m )9 ( p ) ≡ − (cid:18) m + 25 (cid:19) B p − − m ( m + 126 m + 1869 m + 3044) B p − (mod p ) ,R ( m )10 ( p ) ≡ − m ( m + 71 m + 540)( m − (cid:0) B p − B p − + 21 B p − (cid:1) (mod p ) . This conjecture is consistent with the general philosophy we have observed for the finitemultiple zeta values (FMZVs). See, for example, [18, 20] for the definition of FMZVs andthe relevant results. Note that according to the dimension conjecture of FMZVs discoveredby Zagier and independently by the last author (see [20]) the weight 8 (resp. weight 10)piece of FMZVs has conjectural dimension 2 (resp. 3). Theorem 1.1 (i), Conjecture 5.1 andall the previous works in lower weights imply that R ( m ) d ( p ) ( d ≤ 10 and m ≥ 2) should liein the proper subalgebra generated by the so-called A -Bernoulli numbers defined in [20].According to the analogy between FMZVs and MZVs, this subalgebra is the FMZV analogof the MZV subalgebra generated by the Riemann zeta values. It would be interesting to seeif this phenomenon holds in every weight. Acknowledgements. JZ is partially supported by the NSF grant DMS 1162116. Part ofthis work was done while he was visiting the Max Planck Institute for Mathematics, IHESand ICMAT at Madrid, Spain, whose supports are gratefully acknowledged. The authors alsothank the anonymous referee for a number of valuable suggestions which improved the papergreatly. ∗ , KEVIN THIELEN ∗ , LIUQUAN WANG † , AND JIANQIANG ZHAO ⋆ References [1] J. Bl¨umlein, Harmonic sums and Mellin transforms, Nucl. Phys. Proc. Suppl. (1999), pp. 166–168.[2] J. Bl¨umlein, Relations between harmonic sums in massless QCD calculations, Few-Body Systems (2005), pp. 29–34.[3] A. Devoto and D.W. Duke, Table of integrals and formulae for Feynman diagram calculations, Riv.Nuovo Cim. (6) (1984), pp. 1–39.[4] Kh. Hessami Pilehrood, T. Hessami Pilehrood, and R. Tauraso, New properties of multiple harmonicsums modulo p and p -analogues of Leshchiner’s series, Trans. Amer. Math. Soc. , 366 (6) (2014), pp.3131–3159.[5] M.E. Hoffman, Algebraic aspects of multiple zeta values, in: Zeta Functions, Topology and QuantumPhysics, T. Aoki et. al. (eds.), Dev. Math. , Springer, New York, 2005, pp. 51–74.[6] C. Ji, A simple proof of a curious congruence by Zhao, Proc. Amer. Math. Soc. (2005), pp. 3469–3472.[7] M. Kaneko and D. Zagier, Finite multiple zeta values, in preparation.[8] M. Petkovsek, H. Wilf and D. Zeilberger, A=B, A K Peters/CRC Press, 1996.[9] Z. Shen and T. Cai, Congruences for alternating triple harmonic sums, Acta Math. Sinica (Chin. Ser.) , (4) (2012), pp. 737–748.[10] J. A. M. Vermaseren, Harmonic sums, Mellin transforms and integrals, Internat. J. Modern Phys. A (13) (1999), pp. 2037–2076.[11] L. Wang and T. Cai, A curious congruence modulo prime powers, J. Number Theory , (2014), pp.15–24.[12] L. Wang, A curious congruence involving alternating harmonic sums, J. Comb. Number Theory (2014),pp. 209–214.[13] L. Wang, A new curious congruence involving multiple harmonic sums, J. Number Theory (2015),pp. 16–31.[14] L. Wang, A new congruence on multiple harmonic sums and Bernoulli numbers. arXiv:1504.03227.[15] B. Xia and T. Cai, Bernoulli numbers and congruences for harmonic sums, Int. J. Number Theory (4)(2010), pp. 849–855.[16] J. Zhao, Bernoulli numbers, Wolstenholme’s Theorem, and p variations of Lucas’ Theorem, J. NumberTheory (2007), pp. 18–26.[17] J. Zhao, Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory (1)(2008),pp. 73–106.[18] J. Zhao, Mod p structure of alternating and non-alternating multiple harmonic sums. J. Th´eor. NombresBordeaux (1) (2011), pp. 259–268. ( MR Proc. Amer. Math.Soc. (2007), pp. 1329–1333. Email address : [email protected] Email address : [email protected] ∗ Department of Mathematics, Eckerd College, St. Petersburg, FL 33711, USA † Department of Mathematics, National University of Singapore, Singapore, 119076, Sin-gapore ⋆⋆