NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS KATHRIN BRINGMANN AND BEN KANE

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1 NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS KATHRIN BRINGMANN AND BEN KANE To George Andrews, who has been a great inspiration, on the occasion of his 70th birthday Abstract. In previous work, the authors discovered new examples of q-hypergeometric series related to the arithmetic of Q and Q 3. Building on this work, we construct in this paper sum of the tails identities for which some which some of these functions occur as error terms. As an application, we obtain formulas for the generating function of a certain zeta functions for real quadratic fields at negative integers.. Introduction Andrews, Dyson and Hickerson [] investigated the function σq := + n q nn+ = + q q + q q q n which first appeard in Ramanujan s lost notebook [7]. We denote as usual a n := a; q n := n m=0 aqm. By showing that the Fourier coefficients of σq are connected to the arithmetic of Q 6, the authors of [] were able to prove that σ is lacunary, i.e. its coefficents are almost always zero, and yet attains every integer infinitely many times. Since Andrews, Dyson and Hickerson s investigation, the function σq has shown up in a variety of settings and is related to the theory of automorphic forms in many interesting ways, as described in [4]. Here we consider one such type of result involving sum of tails identities. Such identities were first studied by Zagier [8] in his work on Vassiliev invariants, who showed that q q n = q D q + E q, where D q := + n dnq n, E q := n nq n 4. n Here dn denotes the number of divisors of n and n is the Kronecker symbol. In [3], Andrews, Jimenez-Urroz, and Ono obtained a number of related identites, which have the form. F q F n q = F qdq + Eq, where F is a modular infinite product, F n F, and D is a divisor function. Since the coefficients of E grow much slower than those of F D, the function E may be considered as an 000 Mathematics Subject Classification. P8, E6, 05A7. Key words and phrases. q-hypergeometric series, sum of tails, modular form, real quadratic fields. The first author was partially supported by NSF grant DMS

2 KATHRIN BRINGMANN AND BEN KANE error series. Of particular interest are special cases of. in which the error series is σq, for example the following formula which can be found in Ramanujan s lost notebook. q q n = q D q + σq. In [4], the authors discovered 8 additional examples of q-hypergeometric series related to the arithmetic of Q and Q 3 including the functions fq := q n q n+ q n, hq := q n+ q n+ q n+. Using the theory of Bailey pairs, it was shown in [4] that qfq = 4.3 q N a, N a.4 hq = N a q N a, where K := Q, O K is the ring of integers of K, the sum runs over all ideals of O K, and N denotes the norm of an ideal. As in the case of σ, the identities in.3 and.4 imply arithmetic information about the coefficients of f and h including lacunary behavior. In this note we consider sums of tails identities resembling. which involve the functions f and h as error series. To state our results, for i {0, } we define θ i q := n Z q n+i q A q B q A B, we let its finite com- and, denoting as usual the q-binomial coefficients by [ ] A B := panion be given by [ n + θ i,n q := q n n j ] q j+i. We note that lim n θ i,n q = θ i q and that the theta series θ i q are modular forms which can be expressed as an infinite product, namely they can be written θ 0 q and θ q = q q6 ;q 6 q 8 ;q 8. Moreover, we define D q := d o nq n, n= where d 0 n counts the number of odd divisors of n. = q ;q 5 q q 4 ;q 4 Theorem.. We have the sum of tails identities θ 0 q θ 0,n q θ 0 q.5 q q; q q n q; q = n+ q q; q D q hq, θ q θ,n q θ q.6 q q; q q n q; q = n+ q q; q D q + q fq.

3 NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS 3 As an application, we next consider formulas for the generating function of the following zeta functions for real quadratic fields at negative integers, paralleling the results obtained in [8] and [3]. For this define the usual Hecke L-function ζ K s for K and a twisted zeta-function ζ K s by ζ K s := 4 N a s, ζk s := N a s. N a Theorem. will give us the following formulas from which ζ K s and ζ K s may be calculated at negative integers. Theorem.. As a power series in t, we have the equations.7.8 θ 0,n e t e t ; e t n e t ; e t = n ζ K n tn n+ n!, n θ,n e t e t ; e t n e t ; e t = n ζk n tn n+ n!. n We note that similarly we could also consider the limiting behavior at other odds roots of unity. This paper is organized as follows. In Section, we first introduce an auxilliary function with an extra parameter, which is in special cases related to f and h, and thus also to the arithmetic of K. This auxiliary function also has some nice combinatorial meaning. We then show that this function is the error series in a sum of tails identity. In Section 3, we use the results from Section to prove Theorem.. We conclude section 3 with another sums of tails result involving h, but where hq is not the error series. Finally, we will establish the formulas for ζ K s and ζ K s at negative integers in Section 4. Acknowledgements The authors thank George Andrews and Jeremy Lovejoy for fruitful conversations. Moreover, they thank Ken Ono for helpful comments on an earlier version of the paper.. An Auxilliary Function We will first investigate the properties of an auxilliary function which we will then bootstrap to obtain the main results for f and h. To this end, we define g z q := q n q n+ z n q n. We will see in equation.6 that z qg z q is the generating function for a natural combinatorial object. For this recall that an overpartition of the integer n is a partition, where the last occurence of each part may be overlined. Denote by P the set of overpartitions into odd parts where the largest part must be overlined. If we denote the largest part by LΛ and the number of parts by MΛ, then the coefficient of z m q n in z qg z q is the number of overpartitions Λ P of n with MΛ = m, weighted by 4 LΛ. We will particularly be interested in the two specializations z = ±. For the specialization z =, the n-th Fourier coefficient of qg + q equals #{Λ P : LΛ mod 4} #{Λ P : LΛ 3 mod 4},

4 4 KATHRIN BRINGMANN AND BEN KANE while the n-th coefficient of qg q equals #{Λ P : LΛ MΛ + mod 4} #{Λ P : LΛ MΛ + mod 4}. However, we will see that the coefficients of qg ± q will always be equal up to absolute value, due to the following relations with fq and hq: g +q + g q = fq, g +q g q = q hq. Taking q q and adding the two above equations relates the coefficients of g q to the arithmetic of O K by. qg + q = qfq + hq = [ ] 4 + N a i N a q N a, N a which follows from equations.3 and.4. Here we have used the fact that there is a unique ideal of O K of norm, so hq is precisely the generating function for the elements of norm n, weighted by i n. Subtracting the two equations yields. qg q = qfq hq = [ ] 4 + N a + i N a q N a. N a Remark. Since 4 n = 0 and + n+ = 0, it follows that the absolute value of the n-th Fourier coefficient of both qg + q and qg q equals the number of ideals of O K of norm n. Hence, although the coefficients of these series are not multiplicative, they are multiplicative up to ±. We next show a sum of the tail-identity involving the functions g z. For this, we let Theorem.. We have zq.3 zq n zq zq n+ zq n D z q := z q n+. n z q n+ zq n This immediately implies the following corollary. Corollary.. The following equations hold q q n.4 q q n+ q q n.5 q q n+ = zq zq D z q + D q z q gz q. = q q D q q g q, = q q D q + q g q. Proof. From Theorem of [3] with a = zq and t = zq, one may easily deduce that zq zq n = zq D n zq zq n zq z q + D q + + z q n n q n. n z q 3 n

5 NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS 5 Dividing both sides by zq = zq z q and shifting the sum of tails gives zq zq n = zq D zq zq n+ zq z q + D q z q z q n n q n. z q n+ Using Heine s second transformation cf. [, p.,.5] with a = b = q, c = q, and z = z q gives that.6 g z q = z q n z q n+ n q n. This gives the claim. 3. Proof of the Main Result In this section, we will use Theorem. to obtain the desired sum of tails results for fq and hq. Since the proof of both identities are quite similar, we only prove the first here. We first define and its finite companion F q := q + q q q n F n q := q n + q. q n+ q n+ Specializing equation.3 once with z = and once z = and then summing gives F q F n q = F qd q hq. Thus it remains to compute F q and F n q. Using the Jacobi triple product identity cf. [5, p. 7] one obtains F q q = q ; q 4 q ; q q ; q + q ; q 5 q 4 ; q 4 q = q ; q 4 q ; q n Z n q n + q n. n Z Thus To compute F n q we first rewrite F n q = q; q n+ F q = q; q q θ 0 q. q n qn+ + q + q n+ n

6 6 KATHRIN BRINGMANN AND BEN KANE and then use McMahon s finite version of the Jacobi triple product identity [6, vol., Section 33] with x = q for the first summand and x = q for the second summand. This gives F n q = [ ] q; q q n+ n [ ] n+ n + j j q j + + q n+ n q j n + j = [ ] n [ ] q; q q n+ n + j j + q n+ n q n + j + j +j+ = = q; q n+ q; q n+ q n q j q n+j+ + q n+j+ q n j q n+j+ q n j [ ] n + q n j j = q; q θ 0,n τ. n+ q n There is additionally another sum of tails identity related to g z q. In this case g z q does not play the role of the error series but rather as part of the divisor function. Theorem 3.. We have the sum of tails identity q q n q q n q n+ = q q D q hq. Proof. From Theorem of [3] with a = q, b = z q, and c = z q 3, one obtains q q n = q q n q q n z q n+ q q n q n q n q n z n q n n n = q D q z q gz q. q The result then follows after summing the terms with z = and z = and dividing by. In addition to these sum of tails identities, the auxilliary function g z q also allows us to deduce new combinatorial interpretations for fq and hq in terms of overpartitions. Using Heine s first transformation cf. [, p. 0,.] with a = b = q, c = q, and z = z q gives g z q = q q n q q n z q n+ qn. After pulling the infinite product inside the sum, this immediately gives fq = q n+ n q n+ q n+ q n, hq = q n+ n q n+ q n+ q n+. To describe the new combinatorial interpretation for the coefficients of f and h, let n = sλ denote the smallest part of the overpartition Λ. Then fq is the generating function for overpartitions where sλ is overlined and occurs exactly once and sλ+ cannot be overlined, weighted by MΛ.

7 NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS 7 Similarly, writing n + = n + n +, we see that hq is the generating function for overpartitions for which sλ is overlined and occurs precisely once, a non-overlined part of size sλ + must occur, and sλ + cannot be overlined, again weighted by MΛ. 4. Values of zeta functions at negative integers We will use Theorem. to establish Theorem. here. Proof of Theorem.. This proof will follow analogously to that of Zagier [8, Theorem 3] so we will give a brief argument for.7 and leave the details of.8 to the reader. We first note that θ 0 q q q; q = q q ; q 3 q 4 ; q 4 and hence vanishes to infinite order as q. We next replace q by e t and define an and bn by the asymptotic expansions as t 0 given by θ 0,n e t e t ; e t n e t ; e t = ant n, n+ h e t := bnt n. e tn a By Theorem. we have an = bn for every n 0. We then integrate to obtain h e t t s dt = e tn a t s dt = Γs N a s = Γsζ K s. 0 0 The residue at s = n is hence given by n n! ζ K n. However, for any N fixed we have h e t t s dt = bnt n + Ot N t s dt = bn s + n + Hs, n<n 0 n<n where Hs is holomorphic for Res > N, so that for n < N the residue at s = n is bn. References [] G. Andrews, q-series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, CBMS Regional Conference Series in Mathematics, no. 66, Amer. Math. Soc., Providence, RI, pp. 30, 986. [] G. Andrews, F. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math , pages [3] G. Andrews, J. Jimenez-Urroz, K. Ono, q-series identities and values of certain L-functions, Duke Math. J , pages [4] K. Bringmann, B. Kane, Multiplicative q-hypergeometric series arising from real quadratic fields, preprint, arxiv: [5] K. Ono, Web of modularity: Arithmetic of the coefficients of modular forms and q-series, CBMS Regional Conference Series in Mathematics, no. 0, Amer. Math. Soc., Providence, RI, 003. [6] P. A. MacMahon, Combinatory Analysis vol. I, II. Cambridge Univ. Press, 95. Reprinted by Chelsea, New York, 960. [7] S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 988. [8] D. Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology 40 00, pages

8 8 KATHRIN BRINGMANN AND BEN KANE Mathematical Institute, University of Cologne, Weyertal 86-90, 5093 Cologne, Germany address: Math Department, Radboud University, Postbus 900, 6500 GL, Nijmegen, Netherlands address:

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