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  • q-SERIES IDENTITIES AND VALUES OF CERTAIN L-FUNCTIONS

    George E. Andrews, Jorge Jimenez-Urroz and Ken Ono

    Appearing in the Duke Mathematical Journal.

    1. Introduction and Statement of Results.

    As usual, define Dedekinds eta function (z) by the infinite product

    (1.1) (z) := q1/24

    n=1

    (1 qn). (q := e2iz throughout)

    In a recent paper, Zagier [Z; Theorem 2] proved that (note. empty products equal 1 through-out)

    (1.2)

    n=0

    ((24z) q(1 q24)(1 q48) (1 q24n)

    )= (24z)D(q) + E(q)

    where the series D(q) and E(q) are defined by

    D(q) = 12

    +

    n=1

    q24n

    1 q24n= 1

    2+

    n=1

    d(n)q24n = 12

    + q24 + 2q48 + 2q72 + 3q96 + . . . ,

    E(q) =12

    n=1

    (12n

    )nqn

    2=

    12q 5

    2q25 7

    2q49 +

    112q121 + . . . .

    Here d(n) denotes the number of positive divisors of n. This identity plays an importantrole in Zagiers work on Vassiliev invariants in knot theory [Z].

    Two other similar identities were known, and they were noticed by the first author inconnection with one of Ramanujans mock theta functions. In [A2], the first author provedthat

    (1.3)

    n=0

    ((48z)(24z)

    q(1 + q24)(1 + q48) (1 + q24n))

    =(48z)(24z)

    D(q) +M1(q)

    2,

    The first and third authors thank the National Science Foundation for its generous research support. The

    second author thanks PB98-0067 for their support. The third author thanks the Alfred P. Sloan Foundation

    and the David and Lucile Packard Foundation for their support.

    Typeset by AMS-TEX1

  • 2 GEORGE E. ANDREWS, JORGE JIMENEZ-URROZ AND KEN ONO

    (1.4)

    n=0

    ((48z)(24z)

    q(1 q24)(1 q72) (1 q24(2n+1))

    )=(48z)(24z)

    D(q2) +M1(q)

    2

    where M1(q) is the mock theta function given by

    (1.5) M1(q) = q +

    n=1

    q12n2+12n+1

    (1 + q24)(1 + q48) (1 + q24n)= q + q25 q49 + 2q73 .

    The q-series of the function M1(q) was the focus of two extensive studies [A-D-H, C]. Al-though M1(q) is not the Fourier expansion of a modular form, these works show that thecoefficients of M1(q) are given by a Hecke character for the quadratic field Q(

    6). In par-

    ticular, M1(q) enjoys nice properties that one expects for certain weight 1 cusp forms. Forthese reasons, we shall refer to M1(q) and M2(q) (defined in (1.8)) as mock theta functionsalthough they do not exactly fit Ramanujans original definition [A3; p. 291].

    In view of identities (1.2-4), it is natural to investigate the general behavior of q-serieswhich are obtained by summing the iterated differences between an infinite product and itstruncated products. Here we establish two general theorems which yield infinitely manysuch identities, and we illustrate how such identities are useful in determining the values atnegative integers for certain L-functions.

    We shall employ the standard notation

    (1.6) (A; q)n =

    j=0

    (1Aqj)(1Aqn+j)

    ,

    and throughout we assume that |q| < 1 and that the other parameters are restricted todomains that do not contain any singularities of the series or products under consideration.

    Theorem 1.

    n=0

    ((t; q)(a; q)

    (t; q)n(a; q)n

    )

    =

    n=1

    (q/a; q)n(a/t)n

    (q/t; q)n

    +(t; q)(a; q)

    ( n=1

    qn

    1 qn+

    n=1

    qnt1

    1 qnt1

    n=0

    tqn

    1 tqn

    n=0

    aqnt1

    1 aqnt1

    ).

    Theorem 2.

    n=0

    ((a; q)(b; q)(q; q)(c; q)

    (a; q)n(b; q)n(q; q)n(c; q)n

    )

    =(b; q)(a; q)(c; q)(q; q)

    ( n=1

    qn

    1 qn

    n=0

    aqn

    1 aqn

    n=1

    (c/b; q)nbn

    (a; q)n(1 qn)

    ).

  • q-SERIES IDENTITIES 3

    Many interesting specializations of these two theorems yield identities for modular formsthat are eta-products (including identities (1.2-4)). Here we highlight ten of these identities.First we fix notation. We let

    be the operator defined by

    (1.7)

    ( n=0

    a(n)qn)

    =

    n=0

    na(n)qn.

    It is easy to see that the series E(q) in (1.2) is given by

    E(q) =

    ((24z)) /2.

    In addition to the mock theta function M1(q), we shall require the mock theta functionM2(q) defined by

    (1.8) M2(q) =

    n=1

    (1)nq24n21

    (1 q24)(1 q72) (1 q24(2n1))= q23 q47 .

    See [A-D-H] for a detailed study of this function.The ten eta-products F1(z), F2(z), . . . , F10(z) we consider are of the form

    Fi(z) = ai(iz)bi(2iz)

    with ai 6= 0. Obviously, each Fi(z) is a modular form of weight (ai + bi)/2. For each Fi(z)we define quantities ci and fi(j), which are not necessarily unique, for which

    (1.9) Fi(z) = ci

    j=1

    fi(j).

    These are listed in the table below.

    Table 1.

    i Fi(z) i ci fi(j)1 1/(24z) 24 q1 1/(1 q24j)2 (2z)/2(z) 1 1 (1 + qj)/(1 qj)3 (8z)/2(16z) 8 q1 (1 q16j8)/(1 q16j)4 (48z)/(24z) 24 q 1 + q24j

    5 (48z)/(24z) 24 q/(1 q24) 1/(1 q24(2j+1))6 (24z)/(48z) 24 q1 1/(1 + q24j)7 (24z)/(48z) 24 q1 1 q24(2j1)8 (24z) 24 q 1 q24j9 2(z)/(2z) 1 1 (1 qj)/(1 + qj)10 2(16z)/(8z) 8 q/(1 q8) (1 q16j)/(1 q16j+8)

  • 4 GEORGE E. ANDREWS, JORGE JIMENEZ-URROZ AND KEN ONO

    If {1, 8, 24}, then let d(n) be the divisor function defined by

    (1.10) d(n) =

    d(n) =

    d|n 1 if = 24,

    d|n(1)d if = 8,d|n odd 1 if = 1.

    Also, for each i define i by

    (1.11) i ={ 12 if (ai + 2bi)i = 24,

    0 otherwise.

    Notice that i = 1/2 if and only if the order of vanishing of Fi(z) at is 1. The lastquantity we require is i which is defined by

    (1.12) i ={

    2 if i = 5, 7,1 otherwise.

    Theorem 3. If 1 i 10, then

    n=0

    Fi(z) ci nj=1

    fi(j)

    = (1 + [1/i])Fi(z)Di(q) +Gi(q)where [] denotes the greatest integer function,

    Di(q) = i +

    n=1

    di(n)qiin

    and

    Gi(q) =

    0 if i = 1, 2, 3,M1(q)/2 if i = 4, 5,2M2(q)/i if i = 6, 7,

    (i + [2/i])

    (Fi(z)) if i = 8, 9, 10.

    The three forms F1(z), F2(z) and F3(z) have weight -1/2 and the four forms F4(z), F5(z),F6(z) and F7(z) have weight 0. The remaining three forms have weight 1/2. The seriesG4(z), G5(z), G6(z) and G7(z) are mock theta functions, whereas G8(q), G9(q) and G10(q)are the half-derivatives of F8(z), F9(z) and F10(z). In other words, the error series Gi(q)in Theorem 3 satisfy

    Fi(z) Gi(q)

    0 if Fi(z) has weight -1/2,Mock Theta function if Fi(z) has weight 0,

    (Fi(z)) if Fi(z) has weight 1/2.

  • q-SERIES IDENTITIES 5

    Although these identities are elegant in their own right, they are also often useful incalculating the values of L-functions at negative integers. In particular, they lead to analogsof the classical result

    t

    et 1= 1 +

    n=1

    (1)n+1(1 n) tn

    (n 1)!,

    where (s) is the Riemann zeta-function. In this direction, Zagier used (1.2) to show that

    (1.13) et/24

    n=0

    (1 et)(1 e2t) (1 ent) = 12

    n=0

    (1/24)n L(12,2n 1) tn

    n!,

    where 12 is the Dirichlet character with modulus 12 defined by

    12(n) :=

    1 if n 1, 11 (mod 12),1 if n 5, 7 (mod 12),0 otherwise.

    Here we illustrate the generality of this phenomenon by proving the following theorems.

    Theorem 4. As a power series in t, we have

    14

    n=0

    (1 et)(1 e2t) (1 ent)(1 + et)(1 + e2t) (1 + ent)

    =

    n=0

    (1)n(4n+1 1) (2n 1) tn

    n!.

    In addition to (s), we shall consider the Dirichlet L-function

    (1.14) L(2, s) :=

    n=1

    2(n)ns

    ,

    where

    (1.15) 2(n) :=

    1 if n 1, 7 (mod 8),1 if n 3, 5 (mod 8),0 otherwise.

    Theorem 5. As a power series in t, we have

    2et/8

    n=0

    (1 e2t)(1 e4t) (1 e2nt)(1 + et)(1 + e3t) (1 + e(2n+1)t)

    =

    n=0

    (1/8)n L(2,2n 1) tn

    n!.

  • 6 GEORGE E. ANDREWS, JORGE JIMENEZ-URROZ AND KEN ONO

    We shall also consider the Hecke L-function

    (1.16) L(, s) =

    n=1

    a(n)ns

    :=

    aZ[

    6]

    (a)Nas

    where is the order 2 character of conductor 4(3 +

    6) on ideals in Z[

    6] defined by

    (1.17) (a) :={iyx(12x

    )if y is even,

    iyx+1(12x

    )if y is odd,

    when a = (x+ y

    6). If 1 r < 48 is an integer, then let Lr(, s) be the partial L-functiondefined by

    (1.18) Lr(, s) :=

    nr (mod 48)

    a(n)ns

    .

    By the orthogonality of the Dirichlet characters modulo 48 and the analytic continuation ofthe associated twists of L(, s), each Lr(, s) has an analytic continuation to C.

    Theorem 6. As a power series in t, we have

    2et/24

    n=0

    (1et)(1e3t) (1e(2n1)t) =

    n=0

    (1/24)n(L23(,n) + L47(,n))tn

    n!.

    Theorem 7. As a power series in t, we have

    2et/24

    n=0

    (1 et)(1 + e2t) (1 + (1)nent)

    =

    n=0

    (1/24)n (L1(,n) L25(,n)) tn

    n!.

    In 2 we recall certain facts about q-series and basic hypergeometric series, and we proveTheorems 1 and 2. In 3 we prove Theorem 3 and in 4 we prove Theorems 4, 5, 6, and 7.In 5 we examine the partition theoretic consequences of the identities for F1(z) and F8(z).In 6 we give a few more identities which are related to eta-products. The most interestingof these is

    n=0

    (1

    2(24z) 1q2(1 q24)2(1 q48)2 (1 q24n)2

    )=

    12(24z)

    n=1

    (d(n) +m(n)) q24n,

    where m(n) denotes the number of middle divisors of n. A divisor is a middle divisor if itlies in the interval [

    n/2,

    2n).

  • q-SERIES IDENTITIES 7

    Acknoweledgements

    The authors thank Scott Ahlgren, Andrew Granville, Don Zagier and the referee for a varietyof useful comments and suggestions.

    2. Preliminaries and important facts.

    T