Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers

Technical Summary: Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers

Problem Statement
The paper addresses the study of Cauchy numbers of the first kind (Gregory coefficients) and the second kind (Nörlund numbers), traditionally defined via the generating functions involving the logarithm. The authors aim to generalize these sequences to higher orders and extend them to a trivariate framework involving multiple polylogarithms. A primary motivation is to investigate the asymptotic behavior (specifically sign patterns) of these generalized coefficients and to utilize them to construct rational approximations for the reciprocals of multiple zeta values (MZVs) and double zeta values. While the irrationality of odd Riemann zeta values has seen recent progress, the irrationality of general MZVs remains largely open.

Methodology
The authors employ a combination of complex analysis, combinatorial methods, and algebraic identities:

1. Contour Integration: Motivated by Schröder's approach, the authors transform the computation of higher-order coefficients into improper integrals using contour integration in the complex plane. By analyzing the residues of functions involving powers of $\frac{z+1}{\log z - \pi i}$, they derive integral representations for the coefficients.
2. WZ Method: To prove the vanishing of specific higher-order Gregory coefficients, the authors utilize the Wilf-Zeilberger (WZ) method to establish telescoping sums in the evaluation of certain integrals.
3. Sturm's Theorem: The analysis of the sign patterns relies on studying the roots of specific polynomials ($H_\ell(x)$) derived from the integrand. The authors use Sturm's theorem to determine the number and distribution of real roots, which dictates the sign changes in the coefficients.
4. Iterated Path Integrals and Shuffle Relations: For the trivariate extension involving multiple polylogarithms, the authors utilize K. T. Chen's theory of iterated path integrals and shuffle relations to express products of polylogarithms and derive functional equations (such as inversion formulas) necessary for analyzing the behavior of these functions near the positive real axis.
5. Induction and Asymptotic Analysis: The proofs for the eventual positivity of coefficients involve inductive arguments on the power of the $(1+x)$ term and detailed asymptotic estimates of the resulting integrals.

Key Contributions and Results

• Higher-Order Gregory Coefficients ($G_n^{(\ell)}$):

  • The authors define $G_n^{(\ell)}$ via the generating function $(\frac{x}{\log(1+x)})^\ell$.
  • Vanishing Property: Contrary to the classical case ($\ell=1$) where all coefficients are non-zero, the paper proves that for every order $\ell \ge 2$, specific coefficients vanish: $G_{\ell+1}^{(\ell)} = 0$ for even $\ell$, and $G_{\ell}^{(\ell)} = 0$ for odd $\ell \ge 3$.
  • Eventual Alternation: The paper proves that for any fixed order $\ell$, the sequence $G_n^{(\ell)}$ is eventually alternating. Specifically, there exists an integer $Gr(\ell)$ such that for all $n \ge Gr(\ell)$, $(-1)^{n-1}G_n^{(\ell)} > 0$. Numerical evidence suggests $Gr(\ell)$ grows roughly by a factor of 3 as $\ell$ increases.

• Higher-Order Nörlund Numbers:

  • The authors define higher-order Nörlund numbers of the first kind ($\tilde{N}_n^{(\ell)}$) and second kind ($N_n^{(\ell)}$).
  • They prove that $\tilde{N}_n^{(\ell)}$ are strictly alternating for all $n \ge 1$.
  • They prove that $N_n^{(\ell)}$ and a more general class $N_{j,\ell}^n$ (involving powers of $(1+x)$) are eventually alternating.

• Trivariate Extension and Conjecture:

  • A trivariate generalization $C_{j;k;\ell}^n$ is defined using multiple polylogarithms $Li_k(x)$.
  • Conjecture 1.3: The authors conjecture that for any fixed $j, \ell \ge 1$ and composition $k$, the coefficients $C_{0;k;\ell}^n$ are eventually negative and $C_{j;k;\ell}^n$ are eventually positive. Furthermore, they conjecture a lower bound involving the reciprocal of the multiple zeta value $\zeta(k)$.

• Verification in Special Cases:

  • The conjecture is proven for the depth-one case (polylogarithms) and depth-two case (double polylogarithms).
  • For these cases, the authors establish that the coefficients are eventually positive and satisfy the specific lower bound inequality: $C_{j;k;\ell}^n > \binom{n+j-1}{n} \frac{1}{\zeta(k)^\ell}$ for sufficiently large $n$.

• Rational Approximations:

  • As a byproduct, the authors derive infinite families of identities expressing the reciprocals of zeta values $\zeta(k)$ and double zeta values $\zeta(a,b)$ as the limit of a sequence of rational numbers $\nu_{n,k}$ (or $\nu_{n,a,b}$) plus an improper integral that vanishes as $n \to \infty$.
  • Specifically, $1/\zeta(k) = \lim_{n \to \infty} \nu_{n,k}$, where $\nu_{n,k}$ is a rational number derived from the coefficients.

Significance and Claims
The paper claims to be the first to incorporate multiple polylogarithm functions into the framework of Cauchy numbers, creating a vast trivariate extension. The significance of the results lies in:

1. Structural Discovery: Revealing that higher-order Gregory coefficients exhibit vanishing behavior and eventual alternation, properties distinct from their classical counterparts.
2. Irrationality Context: Providing a new framework for rational approximations of MZV reciprocals. The authors note that while these approximations are nontrivial, they hope these results may eventually shed light on the irrationality of multiple zeta values, a problem that remains largely untouched except for specific cases like odd Riemann zeta values.
3. Methodological Extension: Demonstrating that techniques involving contour integration and iterated integrals can be successfully applied to higher-depth polylogarithmic structures to derive asymptotic properties of their coefficients.

The authors remain modest regarding the general case of Conjecture 1.3 for arbitrary depth, noting that the implicit nature of the general inversion formulas (Lemma 9.1) may require new approaches beyond the current methods used for depths 1 and 2. They also pose open questions regarding the arithmetic properties (e.g., congruences) of these new coefficients and their potential connections to prime number distribution.