MULTIPLE ZETA VALUES, POLY-BERNOULLI NUMBERS, AND RELATED ZETA...

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  • T. Arakawa and M. Kaneko Nagoya Math. J. Vol. 153 (1999), 189–209

    MULTIPLE ZETA VALUES,

    POLY-BERNOULLI NUMBERS,

    AND RELATED ZETA FUNCTIONS

    TSUNEO ARAKAWA and MASANOBU KANEKO

    Abstract. We study the function

    ζ(k1, . . . , kn−1; s) = ∑

    0

  • 190 T. ARAKAWA AND M. KANEKO

    probably because “nice properties like the functional equation of the Rie-

    mann zeta function do not seem to extend” to ζ(s1, s2, . . . , sn) for n ≥ 2.

    In this paper, we study the single variable function

    ζ(k1, . . . , kn−1; s) = ∑

    0

  • MULTIPLE ZETA VALUES 191

    ζ(k1, k2, . . . , kn−1; s) := ∑

    0

  • 192 T. ARAKAWA AND M. KANEKO

    Proof. (i) is straightforward, and (ii) is derived from (i) and Li1(z) =

    − log(1 − z) by integrating repeatedly.

    Proposition 2. (i) For Re(s) > 1, we have

    ζ(k1, k2, . . . , kn−1; s) = 1

    Γ(s)

    ∫ ∞

    0

    ts−1

    et − 1 · Lik1,k2,...,kn−1(e

    −t)dt.(1)

    (ii) For Re(s) > 0 and integers n ≥ 2, j ≥ 0, we have

    ∫ ∞

    0 ts+j−1Lik1,k2,...,kn−1(e

    −t)dt = Γ(s + j)ζ(k1, . . . , kn−2; s + j + kn−1).

    Proof. To obtain (i), write the sum in the definition of ζ(k1, k2, . . . ,

    kn−1; s) as

    0

  • MULTIPLE ZETA VALUES 193

    Proof. Writing the sum in the definition of ζ(k1, k2, . . . , kn−1; s) as ∞∑

    m1=1

    ∞∑

    m2=m1+1

    · · · ∞∑

    mn=mn−1+1

    , we can prove (i) by the repetition of the same

    calculation as in the proof of Proposition 2 (i). We prove (ii) by induction

    on n. The case of n = 1 is the analytic continuation of the Riemann zeta

    function. Let N be an arbitrary non-negative integer. Write

    1

    et − 1 =

    N∑

    j=0

    Bj j!

    tj−1 + RN (t)

    with

    RN (t) =

    ∞∑

    j=N+1

    Bj j!

    tj−1,

    where Bj are the classical Bernoulli numbers defined by this expansion. By

    Proposition 2 (ii), we have

    ∫ ∞

    0 ts−1Lik1,k2,...,kn−1(e

    −t)

    N∑

    j=0

    Bj j!

    tj−1dt

    =

    N∑

    j=0

    Bj j!

    Γ(s + j − 1)ζ(k1, . . . , kn−2; s + j + kn−1 − 1),

    and thus

    Γ(s)ζ(k1, k2, . . . , kn−1; s) =

    ∫ ∞

    0 ts−1Lik1,k2,...,kn−1(e

    −t)RN (t)dt(2)

    +

    N∑

    j=0

    Bj j!

    Γ(s + j − 1)ζ(k1, . . . , kn−2; s + j + kn−1 − 1).

    The second term in the right-hand side of (2) can be analytically continued

    to a meromorphic function in the whole s-plane by the induction assump-

    tion. Write the integral in (2) as a sum of two integrals:

    ∫ 1

    0 ts−1Lik1,k2,...,kn−1(e

    −t)RN (t)dt +

    ∫ ∞

    1 ts−1Lik1,k2,...,kn−1(e

    −t)RN (t)dt.

  • 194 T. ARAKAWA AND M. KANEKO

    The second integral converges absolutely for any s ∈ C and represents an

    entire function of s. Using the estimates

    Lik1,k2,...,kn−1(e −t) ≤

    

    

    Lik1,k2,...,kn−1(1) = ζ(k1, k2, . . . , kn−1),

    for kn−1 > 1,

    Li1,1,...,1 ︸ ︷︷ ︸

    n−1

    (e−t) = 1

    (n − 1)! (− log(1 − e−t))n−1

    for kn−1 = 1,

    and

    RN (t) = O(|t| N ) (|t| < 1),

    we see that the first integral converges absolutely for Re(s + N) > 0 and

    hence holomorphic as a function of s in the same region. Since we can choose

    N arbitrary large, this shows via (2) that the function ζ(k1, k2, . . . , kn−1; s)

    has a meromorphic continuation to the whole s-plane.

    By means of the inductive relation (2), we can, in principle, detect the

    location and the order of poles of ζ(k1, k2, . . . , kn−1; s), which are in general

    quite complicated to describe. Possible poles lie at the integers ≤ 1. Here

    we only state results at s = 1 as a proposition.

    Proposition 4. (i) Let the index set (k1, k2, . . . , kn−1) be of the form

    (k1, . . . , ki, 1, 1, . . . , 1 ︸ ︷︷ ︸

    m

    ) with m ≥ 0, i ≥ 1, and ki > 1. Then, the order of

    pole of ζ(k1, k2, . . . , kn−1; s) at s = 1 is m + 1 and the leading coefficient is

    ζ(k1, k2, . . . , ki) (i = n − m − 1), i.e.

    ζ(k1, k2, . . . , kn−1; s) = ζ(k1, k2, . . . , ki)

    (s − 1)m+1 + O((s − 1)−m).

    (ii) For (k1, k2, . . . , kn−1) = (1, 1, . . . , 1), we have

    Γ(s)ζ(1, 1, . . . , 1 ︸ ︷︷ ︸

    n−1

    ; s) = 1

    (s − 1)n + O(s − 1).(3)

    Proof. (i) The integral in (2) is holomorphic at s = 1 even when N = 0,

    in which case the relation (2) divided by Γ(s) reads

    ζ(k1, k2, . . . , kn−1; s) = 1

    Γ(s)

    ∫ ∞

    0 ts−1Lik1,k2,...,kn−1(e

    −t)R0(t)dt

    + 1

    s − 1 ζ(k1, . . . , kn−2; s + kn−1 − 1).

  • MULTIPLE ZETA VALUES 195

    By our definition of m, the function ζ(k1, . . . , kn−m−2; s + kn−m−1 − 1) is

    holomorphic at s = 1 with the value ζ(k1, k2, . . . , kn−m−1). From this the

    case of m = 0 follows, and we can easily obtain the proposition by induction

    on the number of indices ki, starting with

    ζ(s) = 1

    s − 1 + O(1).

    (ii) By Proposition 2 (i) and Lemma 1 (ii), we have

    Γ(s)ζ(1, 1, . . . , 1 ︸ ︷︷ ︸

    n−1

    ; s) = (−1)n−1

    (n − 1)!

    ∫ ∞

    0

    ts−1

    et − 1

    ( log

    ( 1 − e−t

    ))n−1 dt.

    Let ζ∗(1, 1, . . . , 1 ︸ ︷︷ ︸

    n−1

    ; s) be defined by

    Γ(s)ζ∗(1, 1, . . . , 1 ︸ ︷︷ ︸

    n−1

    ; s) = (−1)n−1

    (n − 1)!

    ∫ ∞

    0

    ts−1

    et − 1

    (

    log

    ( 1 − e−t

    t

    ))n−1

    dt.

    By the standard method already used to prove the analytic continuation of

    ζ(k1, k2, . . . , kn−1; s), we see that the function ζ ∗(1, 1, . . . , 1

    ︸ ︷︷ ︸

    n−1

    ; s) is continued

    to an entire function in the whole s-plane. Now we claim that the following

    relation holds:

    Γ(s)ζ(1, . . . , 1 ︸ ︷︷ ︸

    n−1

    ; s) = (−1)n−1

    (n − 1)! ·

    dn−1

    dsn−1 (Γ(s)ζ(s))(4)

    +

    n−2∑

    j=0

    (−1)j

    j! ·

    dj

    dsj

     Γ(s)ζ

    ∗(1, . . . , 1 ︸ ︷︷ ︸

    n−1−j

    ; s)

      .

    This can be seen by using

    ( log(1 − e−t)

    )n−1 =

    n−1∑

    j=0

    ( n − 1

    j

    )

    (log t)j (

    log

    ( 1 − e−t

    t

    ))n−1−j

    and ∫

    0

    ts−1

    et − 1 (log t)j

    (

    log

    ( 1 − e−t

    t

    ))n−1−j

    dt

    = (−1)n−1−j(n − 1 − j)! dj

    dsj

    {

    Γ(s)ζ∗(1, . . . , 1 ︸ ︷︷ ︸

    n−1−j

    ; s) }

    .

  • 196 T. ARAKAWA AND M. KANEKO

    Using (4) and the formula

    Γ(s)ζ(s) = 1

    s − 1 + O(s − 1)

    which is equivalent to the more familiar (“Kronecker limit formula for ζ(s)”,

    see e.g. Siegel [S])

    ζ(s) = 1

    s − 1 + γ + O(s − 1) (γ = Euler’s constant),

    we conclude that

    Γ(s)ζ(1, . . . , 1 ︸ ︷︷ ︸

    n−1

    ; s) = 1

    (s − 1)n + O(1).

    Let c (n) 0 be the constant term on the right-hand side. From the relation

    (obtained from (2) by putting N = 0 and ∀ki = 1)

    ζ(1, . . . , 1 ︸ ︷︷ ︸

    n−1

    ; s)(5)

    = 1

    Γ(s)

    ∫ ∞

    0 ts−1Li1,1,...,1

    ︸ ︷︷ ︸

    n−1

    (e−t)R0(t)dt + 1

    s − 1 ζ(1, . . . , 1

    ︸ ︷︷ ︸

    n−2

    ; s),

    we obtain

    1

    (s − 1)n + c

    (n) 0 + O(s − 1) =

    1

    (s − 1)n +

    c (n−1) 0

    s − 1 + O(1),

    hence we necessarily have c (n−1) 0 = 0. Since n is arbitrary, this completes

    the proof of the Proposition.

    Remarks. 1. The values of the holomorphic function ζ∗(1, 1, . . . , 1 ︸ ︷︷ ︸

    r

    , s)

    at non-positive integers are evaluated in terms of Bernoulli numbers:

    ζ∗(1, 1, . . . , 1 ︸ ︷︷ ︸

    r

    , 1 − m) = (−1)r+m−1

    r! · D

    (r) m

    m (m ≥ 1),

    where D (r) m is defined by

    1

    et − 1 ·

    {

    log

    ( 1 − e−t

    t

    )}r

    =

    ∞∑

    m=1

    D(r)m · tm−1

    m! ,

  • MULTIPLE ZETA VALUES 197

    (note that D (r) m = 0 for 1 ≤ m ≤ r − 1). It is easy to see that (noting

    (

    log (

    1−e−t

    t

    )) ′

    = 1 et−1 −

    1 t )

    D(1)m =

    m−1∑

    j=0

    ( m

    j

    )

    · Bj · Bm−j m − j

    and that

    D(r+1)m =

    m−1∑

    l=r

    ( m

    l

    )

    · D (r) l ·

    Bm−l m − l

    for r ≥ 1 and m ≥ r + 1. Hence D (r) m is the sum of (r + 1)-fold products of

    Bernoulli numbers.