The Analytic Structure of the MRB Constant

Abel–Plana Summation, Reciprocal Residues, and a Universal Greedy Expansion

Marvin Ray Burns

Abstract

The MRB constant is defined by the conditionally convergent alternating series
C
MRB

∞
∑
n1
n
(-1)
(
1/n
n
-1)
with numerical value
C
MRB
≈0.1878596424620671202485179340542732300559…
No exact closed form is presently known. This notebook organizes the analytic and computational structure around:
◼
  • alternating Abel–Plana summation,
    • ◼
  • contour deformation,
    • ◼
  • principal-branch logarithmic singularities,
    • ◼
  • reciprocal residues,
    • ◼
  • and the induced greedy approximation system.
    • The main themes are:
    1. a vertical contour representation for
    C
    MRB
    ;
    2. recovery of the original series terms as positive-integer residues;
    3. emergence of the reciprocal branch kernel
    K(u)
    u-2
    u
    sin(πu)
    sin(π/u)
    ;
    4. a residue family
    A(m)
    2
    π
    -1/m
    m
    sin
    π
    m
    ,
    whose
    m8
    term gives a striking approximation to
    C
    MRB
    ;
    5. a signed greedy expansion with asymptotically quadratic error contraction.

    1. The MRB constant

    We begin with the defining alternating series.
    In[]:=
    ClearAll[mrbTerm,mrbPartialSum,mrbN];​​​​mrbTerm[n_Integer?Positive]:=(-1)^n(n^(1/n)-1)​​mrbPartialSum[n_Integer?Positive]:=Sum[mrbTerm[k],{k,1,n}]
    A high-precision numerical reference value can be stored as follows:
    In[]:=
    mrbN=SetPrecision[​​0.1878596424620671202485179340542732300559030949,​​80​​];
    A quick numerical check:
    In[]:=
    Table[{n,N[mrbPartialSum[n],20]},{n,5,30,5}]
    Out[]=
    {{5,0.0064478929775668828917},{10,0.31323175925465597513},{15,0.091199962508263363034},{20,0.26720512193053489545},{25,0.12016286194066093451},{30,0.24713782179889177015}}

    2. Elementary asymptotics and conditional convergence

    Since
    1/n
    n
    exp
    logn
    n
    ,
    we have
    1/n
    n
    -1
    logn
    n
    +O
    2
    log
    n
    2
    n
    ,
    hence
    n
    (-1)
    (
    1/n
    n
    -1)∼
    n
    (-1)
    logn
    n
    .
    This explains conditional convergence.
    Symbolic expansion scaffold:
    In[]:=
    Normal@Series[Exp[Log[n]/n]-1,{n,Infinity,3}]
    Out[]=
    Log[n]
    n
    +
    2
    Log[n]
    2
    2
    n
    +
    3
    Log[n]
    6
    3
    n
    For large-
    n
    numerical comparison:
    In[]:=
    Table[​​{​​n,​​N[n^(1/n)-1,20],​​N[Log[n]/n,20]​​},​​{n,{10,100,1000,10000}}​​]
    Out[]=
    {{10,0.25892541179416721042,0.23025850929940456840},{100,0.047128548050899533465,0.046051701859880913680},{1000,0.0069316688518041699297,0.0069077552789821370521},{10000,0.00092145831929587610817,0.00092103403719761827361}}

    3. The principal-branch analytic function

    Define
    Φ(z)exp
    z
    z
    -1,
    where
    denotes the principal branch.
    In[]:=
    ClearAll[phi];​​phi[z_]:=Exp[Log[z]/z]-1
    This is the core analytic continuation of the summand shape.
    Useful related functions:
    In[]:=
    ClearAll[verticalIntegrand,reciprocalKernel,residueMagnitude];​​verticalIntegrand[z_]:=phi[z]Csc[Piz]​​reciprocalKernel[u_]:=u^(u-2)Sin[Piu]/Sin[Pi/u]​​residueMagnitude[m_Integer?Positive]:=(2/Pi)m^(-1/m)Sin[Pi/m]

    4. Vertical contour representation

    The notebook centers the contour formula
    C
    MRB
    
    i
    2
    1+i∞
    ∫
    1-i∞
    Φ(z)csc(πz)z.
    Under the parametrization
    z1+it
    ,
    dzit
    , this becomes
    C
    MRB
    -
    1
    2
    ∞
    ∫
    -∞
    Φ(1+it)csc(π(1+it))t.

    4.1 Regularized real-line integrand

    The removable point at
    t0
    is handled explicitly.
    In[]:=
    ClearAll[verticalLineIntegrand];​​verticalLineIntegrand[t_?NumericQ]:=​​If[​​t==0,​​1/(2Pi),​​-1/2phi[1+It]Csc[Pi(1+It)]​​]

    4.2 Numerical contour evaluation

    In[]:=
    ClearAll[cmrbVertical];​​cmrbVertical[prec_:50]:=Module[{wp=prec+30},​​NIntegrate[​​verticalLineIntegrand[t],​​{t,-Infinity,0,Infinity},​​WorkingPrecision->wp,​​PrecisionGoal->prec,​​AccuracyGoal->prec,​​Method->"DoubleExponential"​​]​​]
    Comparison with the stored reference value:
    In[]:=
    {​​N[cmrbVertical[40],30],​​N[mrbN,30],​​N[cmrbVertical[40]-mrbN,20]​​}
    Out[]=
    {0.187859642462067120248517934054,0.187859642462067120248517934054,1.3878617201721584187×
    -49
    10
    }

    5. The removable singularity at
    z1

    The contour integrand appears singular at
    z1
    , but this singularity is removable.
    Numerical and symbolic checks:
    and for the parametrized real-line form,

    6. Positive-integer residues recover the MRB summands

    6.1 Symbolic verification

    6.2 Interpretation

    This gives a clean residue-theoretic meaning to the defining series: the positive-integer residues of the contour integrand are exactly the summand data encoded analytically.

    7. Branch cut structure and reciprocal substitution

    8. The reciprocal branch kernel

    8.2 Residues at reciprocal poles

    Direct residue table:

    9. The positive residue-magnitude family

    A small table:
    This family is central both analytically and algorithmically.

    11.1 Symbolic computation

    12.1 Exploratory symbolic expansion

    12.2 Numerical confirmation

    The final column should tend toward 2.

    13. Greedy residue approximation system

    13.1 Heuristic consequence of the asymptotics

    14. Greedy expansion implementation

    This section reproduces the computational scaffold from the paper in notebook form.
    Positive residue magnitude:
    Refined continuous predictor:
    Initial contour-derived approximation:
    Nearest-index search near the predictor:
    Greedy recursion:
    Display table:
    Expected first selected indices:
    and expected absolute residual scales:
    which illustrates near-doubling of correct digits.

    15. The core contraction interval

    This suggests the global strategy:
    ◼
  • after finitely many such steps, the residual enters the core interval;
    • ◼
  • then nearest-residue selection takes over and eventually enters the asymptotically quadratic regime.
    • This is the structural heart of the proposed Universal Greedy Residue Expansion Theorem (UGRET).

    16. UGRET notebook framework

    This section provides a clean theorem-outline scaffold.

    Theorem template

    Proof-outline placeholders

    ◼
  • Step 5: Estimate nearest-neighbor spacing and deduce

    • 17. Numerical experiments beyond MRB

    Because UGRET is presented as universal, the notebook should include experiments for generic targets.
    Suggested targets:

    18. Symbolic and numerical verification appendix

    This section collects the most important checks in one place.

    18.1 Vertical contour value

    18.2 Positive-integer residue identity

    18.4 Reciprocal residue family

    19. Precision and reproducibility notes

    This distinction matters:
    ◼
  • finite-precision zero must not be interpreted as exact vanishing;
    • Recommended practice:

    20. Research directions

    This notebook naturally opens several lines of inquiry.

    Analytic questions

    ◼
  • Can the branch-cut deformation be made globally rigorous with a complete indentation prescription at the negative integers?
    • ◼
  • Is there a direct exact relation between the vertical contour and the reciprocal residue family beyond the currently isolated local mechanism?

    • Asymptotic questions

    ◼
  • Can the $1/4$ constant in the quadratic contraction estimate be sharpened or made effective with explicit bounds?

    • Arithmetic questions

    Computational questions

    ◼
  • How stable is the predictor formula under extremely high precision?

    • 21. Documentation links

    Useful Wolfram Language references:
    Related documentation mentioned by the analytic framework:

    22. Closing summary

    This notebook is designed as a research notebook and verification framework for the analytic structure surrounding the MRB constant.
    It organizes the following chain of ideas:
    ◼
  • the alternating MRB series,
    • ◼
  • its Abel–Plana vertical contour representation,
    • ◼
  • recovery of the MRB summands as residues at positive integers,
    • ◼
  • branch-cut analysis leading to the reciprocal kernel
    • ◼
  • the residue magnitude family
    • ◼
  • and the universal signed greedy expansion principle UGRET.
    • The key conceptual surprise is that a contour-derived residue family appears to support not only a local analytic interpretation but also a globally effective approximation system with asymptotically quadratic convergence.
    If you want, I can next turn this into one of the following:
    1. a paper-style notebook with theorem/proof formatting and polished exposition,
    2. a fully executable computation notebook with evaluated cells and plots,
    3. a seminar notebook with shorter sections and presentation flow,
    4. or a journal-submission outline with abstract, theorem list, appendix structure, and bibliography placeholders.