The Analytic Structure of the MRB Constant

Abel–Plana Summation, Contour Deformation, and an Unexpected Closed-Form ApproximationMarvin Ray Burns

Abstract

The MRB constant is defined by the conditionally convergent alternating series
C
MRB

∞
∑
n1
n
(-1)
(
1/n
n
-1)0.1878596424620671202485179340542732300559…
No exact closed form is presently known.
We study the constant through an alternating Abel–Plana representation and an equivalent vertical contour integral. If
Φ(z)exp
Logz
z
-1,
then
C
MRB

i
2
1+i∞
∫
1-i∞
Φ(z)csc(πz)z.
The apparent singularity at
z1
is removable, and the residues of the integrand at the positive integers reproduce the individual terms of the original MRB series.
Deformation toward the principal logarithmic branch cut on the negative real axis leads naturally to the reciprocal kernel
u-2
u
sin(πu)
sin(π/u)
.
Its residues generate the family
2
m
(-1)
π
-1/m
m
sin
π
m
.
The member corresponding to
m8
produces the striking closed-form approximation
C
MRB
≈
5/8
2
sin(π/8)
π

2-
2
3/8
2
π
,
whose error is approximately
1.89
-8
10
. The approximation is therefore not introduced by fitting the decimal expansion of the constant; it arises from the analytic structure of the contour representation.

1 Introduction

The MRB constant is
C
MRB

∞
∑
n1
n
(-1)
(
1/n
n
-1).
Numerically,
C
MRB
0.1878596424620671202485179340542732300559030949…
Since
1/n
n
exp
logn
n
,
we have
1/n
n
-1
logn
n
+O
2
log
n
2
n
,
and therefore
n
(-1)
(
1/n
n
-1)∼
n
(-1)
logn
n
.
Thus the defining series converges conditionally.
The purpose of the present investigation is not to claim an exact closed form for
C
MRB
, but rather to expose some of the complex analytic structure underlying the constant.
The principal chain developed below is
◼
  • vertical Abel–Plana contour
    • ◼
  • shift right
    ↙
    MRB series
    • ◼
  • shift left
    ↘
    negative-real branch cut
    • ◼
  • branch-cut kernel
    u-2
    u
    sin(πu)
    sin(π/u)
    • ◼
  • exceptional approximation
    2-
    2
    3/8
    2
    π
    • The right-hand path leads to the main numerical surprise of the paper: an exceptionally accurate elementary approximation arising from the
    m8
    member of an analytically derived residue family.

    2 The alternating Abel–Plana representation

    Define
    g(z)
    1/z
    z
    -1exp
    Logz
    z
    -1,
    using the principal branch of the complex logarithm.
    Shift the original series by writing
    f(z)g(z+1)exp
    Log(1+z)
    1+z
    -1.
    Then
    C
    MRB
    -
    ∞
    ∑
    m0
    m
    (-1)
    f(m).
    The alternating Abel–Plana formula gives
    ∞
    ∑
    m0
    m
    (-1)
    f(m)
    1
    2
    f(0)+
    i
    2
    ∞
    ∫
    0
    f(it)-f(-it)
    sinh(πt)
    t.
    Since
    f(0)0,
    we obtain
    C
    MRB
    -
    i
    2
    ∞
    ∫
    0
    exp
    Log(1+it)
    1+it
    -exp
    Log(1-it)
    1-it
    
    sinh(πt)
    t.
    For real
    t
    , the two exponential terms are complex conjugates. Therefore
    C
    MRB
    
    ∞
    ∫
    0
    csch(πt)ℑexp
    Log(1+it)
    1+it
    dt.
    This is the alternating Abel–Plana integral representation used throughout the subsequent analysis.

    3 The vertical contour representation

    Define
    Φ(z)exp
    Logz
    z
    -1.
    Set
    z1+it, dzit.
    Since
    sin(π(1+it))-isinh(πt),
    we have
    csc(π(1+it))
    i
    sinh(πt)
    .
    Combining the positive and negative halves of the Abel–Plana integral therefore yields
    C
    MRB
    
    i
    2
    1+i∞
    ∫
    1-i∞
    Φ(z)csc(πz)z.
    Equivalently,
    C
    MRB
    
    i
    2
    1+i∞
    ∫
    1-i∞
    exp
    Logz
    z
    -1csc(πz)z.
    This vertical representation is particularly useful because its residue structure encodes the original MRB series.

    4 The removable singularity at
    z1

    At first sight,
    Φ(z)csc(πz)
    appears singular at
    z1
    , because
    csc(πz)
    has a simple pole there. However,
    Φ(1)0
    , and the singularity is removable.

    Lemma 1

    The function
    Φ(z)csc(πz)
    extends analytically across
    z1
    , with
    lim
    z1
    Φ(z)csc(πz)-
    1
    π
    .
    Proof. As
    z1
    ,
    Logz(z-1)-
    1
    2
    2
    (z-1)
    +O
    3
    (z-1)
    .
    Hence
    Logz
    z
    (z-1)+O
    2
    (z-1)
    ,
    and therefore
    Φ(z)
    Logz/z
    e
    -1(z-1)+O
    2
    (z-1)
    .
    Also,
    sin(πz)-π(z-1)+O
    3
    (z-1)
    ,
    so
    csc(πz)-
    1
    π(z-1)
    +O(z-1).
    Thus
    Φ(z)csc(πz)-
    1
    π
    +O(z-1),
    which proves the result.

    Remark 1

    The removable singularity is relevant in numerical quadrature. Evaluating
    Φ(1+it)csc(π(1+it))
    as a product of separate factors can lead to numerical indeterminacy at
    t0
    , even though the product has a finite limit.
    After including the factor arising from the parametrized vertical contour, the real
    t
    -integrand has limiting value
    1
    2π
    at
    t0
    .

    5 Positive-integer residues and recovery of the MRB series

    The positive-integer poles of
    csc(πz)
    encode the original series exactly.

    Theorem 1

    For every integer
    n≥2
    ,
    *
    Res
    zn
    [Φ(z)csc(πz)]
    n
    (-1)
    π
    (
    1/n
    n
    -1).
    Proof. At an integer
    n
    ,
    *
    Res
    zn
    csc(πz)
    n
    (-1)
    π
    .
    Since
    Φ(n)exp
    logn
    n
    -1
    1/n
    n
    -1,
    multiplication gives
    *
    Res
    zn
    [Φ(z)csc(πz)]
    n
    (-1)
    π
    (
    1/n
    n
    -1).
    Thus, apart from the universal factor
    1/π
    , the residues of the vertical-contour integrand are precisely the terms of the MRB series.
    Symbolically,
    vertical contour shifted right
    ⟹
    MRB series.

    6 The principal logarithmic branch cut

    The non-meromorphic behavior of
    Φ(z)exp
    Logz
    z
    -1
    comes from the principal logarithm.
    We use
    -π<argz≤π,
    so the branch cut of
    Logz
    lies along
    (-∞,0].
    For
    z-x, x>0,
    the two boundary values are
    Log(-x+i0)logx+iπ,
    and
    Log(-x-i0)logx-iπ.
    Therefore
    Φ(-x+i0)exp-
    logx+iπ
    x
    -1
    -1/x
    x
    -iπ/x
    e
    -1,
    while
    Φ(-x-i0)
    -1/x
    x
    iπ/x
    e
    -1.
    Their discontinuity is therefore
    DiscΦ(-x)Φ(-x+i0)-Φ(-x-i0)-2i
    -1/x
    x
    sin
    π
    x
    .
    Thus the complicated logarithmic expression has an unexpectedly simple jump across its branch cut.

    7 The reciprocal branch kernel

    Along the negative real axis,
    csc(-πx)-csc(πx).
    Consequently the branch discontinuity naturally produces the real kernel
    -1/x
    x
    sin(π/x)
    sin(πx)
    .
    Now make the reciprocal substitution
    u
    1
    x
    , x
    1
    u
    , dx-
    du
    2
    u
    .
    Since
    -1/x
    x
    
    u
    u
    ,
    the transformed kernel becomes
    K(u)
    u-2
    u
    sin(πu)
    sin(π/u)
    .
    Thus the reciprocal kernel is not introduced by numerical fitting. It arises directly from the logarithmic discontinuity of the vertical Abel–Plana contour.

    8 Singularities of the reciprocal kernel

    The denominator vanishes when
    sin
    π
    u
    0,
    or equivalently
    1
    u
    m, m∈Z.
    For positive
    u
    , the relevant points are
    u
    1
    m
    , m1,2,3,…
    At
    u1
    , numerator and denominator vanish simultaneously. Indeed,
    lim
    u1
    K(u)-1,
    so
    u1
    is removable.
    The remaining points
    u
    1
    2
    ,
    1
    3
    ,
    1
    4
    ,…
    are simple poles.

    9 Residues of the reciprocal kernel

    Theorem 2

    For every integer
    m≥2
    ,
    *
    Res
    u1/m
    K(u)
    m+1
    (-1)
    π
    -1/m
    m
    sin
    π
    m
    .
    Proof. Write
    K(u)
    u-2
    u
    sin(πu)
    sin(π/u)
    .
    The denominator has a simple zero at
    u1/m
    , and
    d
    du
    sin
    π
    u
    -
    π
    2
    u
    cos
    π
    u
    .
    Therefore
    d
    du
    sin
    π
    u
    u1/m
    -π
    2
    m
    m
    (-1)
    .
    Also,
    1/m-2
    1
    m
    
    2-1/m
    m
    .
    Hence
    *
    Res
    u1/m
    K(u)
    2-1/m
    m
    sin(π/m)
    -π
    2
    m
    m
    (-1)
    
    m+1
    (-1)
    π
    -1/m
    m
    sin
    π
    m
    .
    It is convenient to define the associated doubled residue family
    R
    m
    -2*
    Res
    u1/m
    K(u)
    2
    m
    (-1)
    π
    -1/m
    m
    sin
    π
    m
    .

    10 The exceptional case
    m8

    Its interest lies not merely in its numerical accuracy, but in its analytic origin: the expression is obtained from a specific member of a residue family produced by the branch structure of the contour.

    12 A general finite-part formula

    13 Differential structure of the branch discontinuity

    14 Structure of the local coefficients

    At present, the observed coefficient structure is best regarded as a local algebraic pattern; it does not by itself imply that the global branch contribution belongs to the same algebra.

    15 Interpretation of the approximation

    The central analytic chain may be summarized as
    ◼
  • Vertical Abel–Plana contour

    • 16 Greedy approximation within the residue family

    16.1 Asymptotic prediction of the indices

    16.2 Quadratic decay of the residual

    16.3 A conjectural greedy residue expansion

    The numerical evidence indicates approximately quadratic convergence, but no proof of convergence of this greedy construction is presently known.

    17 What remains unresolved

    Any exact contour decomposition must retain the contour prescription, including the branch cut, orientations, and pole indentations.
    This is important: the reciprocal kernel is an exact local object arising from contour deformation, but the complete global contour identity requires careful treatment of these singularities before an exact closed-form evaluation can be claimed.

    18 Open problems

    The analysis suggests several natural problems.
    1. Can the complete negative-real-axis branch contribution be evaluated in closed form?
    2. Can a finite keyhole-contour identity be derived that cleanly separates the branch contribution and the pole indentation terms before the limit to infinity is taken?
    4. Does the greedy residue construction
    5. Does the reciprocal kernel possess a useful functional equation under
    8. Can the asymptotic index law

    19 Conclusion

    The investigation does not produce an exact closed form for the MRB constant. It does, however, reveal a coherent complex-analytic structure behind its defining alternating series.

    Acknowledgments

    This investigation developed through a combination of analytic calculation, high-precision numerical experimentation, and symbolic verification.
    ChatGPT was used interactively to explore possible contour representations, branch-cut transformations, residue formulas, and related analytic questions. The formulas used in the manuscript were subsequently tested numerically or symbolically in Mathematica where appropriate.

    Appendix A. Selected Mathematica checks

    This appendix contains representative Mathematica computations used to verify the principal symbolic identities and numerical evaluations appearing in the paper. The programs are intended as independent checks of the analytic derivations rather than as proofs.

    (A) Numerical verification of the vertical contour representation

    (B) Symbolic verification of the residue formula

    (C) Verification of the removable singularity of the reciprocal kernel

    which returns

    (D) Symbolic verification of the residue family

    (F) Numerical construction of the greedy residue approximation