Deficiency Identity for Zeta Values and Optimal Approximation

1. Problem Statement

The Riemann zeta function, $\zeta(s) = \sum_{n=1}^\infty n^{-s}$, is a fundamental object in analytic number theory. While explicit closed-form expressions exist for even integers (involving Bernoulli numbers), no such expressions are known for odd integers ($\zeta(3), \zeta(5), \dots$).
Standard numerical approximation methods include:

• Dirichlet Truncation: Summing the first $n$ terms. This yields an error of $O(n^{-(q-1)})$.
• Euler–Maclaurin Summation: Expanding the tail of the series asymptotically to achieve higher-order convergence. While highly effective, this requires explicit calculation of Bernoulli numbers and higher-order derivatives, increasing symbolic complexity.

The paper addresses the challenge of approximating odd zeta values $\zeta(q)$ with high efficiency and a unified structural framework, without relying on explicit tail expansions or complex symbolic machinery.

2. Methodology: The Deficiency Framework

The core innovation is a deficiency-based representation derived from the convexity of nonlinear transformations.

2.1. Core Concept

The method relies on the observation that for a convex function $f(x) = x^\alpha$ ($\alpha > 1$) and positive increments, aggregating quantities before applying the transformation yields a larger value than applying the transformation locally and summing.

• Global Aggregation: $(\sum k^{-p})^{q/p}$
• Local Accumulation: $\sum (k^{-p})^{q/p} = \sum k^{-q}$

The difference between these two quantities defines the Deficiency Functional.

2.2. Exact Identity

Let $S_n^{(p)} = \sum_{k=1}^n k^{-p}$ and $S_\infty^{(p)} = \zeta(p)$. For $q > p$, the paper defines the cumulative deficiency $D_{p,q}^n$ such that:
$$\zeta(q) = \zeta(p)^{q/p} - D_{p,q}^\infty$$
where $D_{p,q}^\infty$ represents the "loss" incurred by fragmenting the mass before applying the nonlinear transformation.
Specifically, the finite deficiency is defined as:
$$D_{p,q}^n = (S_n^{(p)})^{q/p} - \sum_{k=1}^n k^{-q}$$
This yields the exact decomposition:
$$\zeta(q) = (S_n^{(p)})^{q/p} - D_{p,q}^n$$
This structure allows the approximation of $\zeta(q)$ by calculating the partial sum of the base sequence $S_n^{(p)}$, raising it to the power $q/p$, and subtracting the accumulated deficiency.

2.3. Approximation Scheme and Bias Correction

The authors propose a corrected estimator $B_{p,q}^n$ to handle the bias introduced by the tail behavior of the base sequence.

1. Basic Estimator ($A_{p,q}^n$): Uses the partial deficiency directly. It suffers from a first-order bias of $O(n^{-1})$.
2. Corrected Estimator ($B_{p,q}^n$): By subtracting an explicit first-order bias term derived from the Taylor expansion of the nonlinear transform, the method achieves higher convergence.
   $$B_{p,q}^n = \zeta(p)^{q/p} - D_{p,q}^n - \frac{q}{p}\zeta(p)^{q/p-1}(\zeta(p) - S_n^{(p)})$$

3. Key Contributions

• Exact Structural Decomposition: The paper establishes a rigorous identity expressing $\zeta(q)$ as a nonlinear aggregate of a base sequence minus a cumulative deficiency. This provides a new geometric and algebraic perspective on zeta values.
• Unified Approximation Framework: Unlike Euler–Maclaurin, which requires specific expansions for each $q$, this method uses a single base sequence $S_n^{(p)}$ (typically $p=2$ or optimized $p$) to approximate any $q > p$.
• Implicit Euler–Maclaurin Structure: The authors demonstrate that the bias correction terms arise naturally from the Taylor expansion of the nonlinear map $x \mapsto x^{q/p}$. Thus, the method implicitly generates asymptotic corrections similar to Euler–Maclaurin without explicitly computing Bernoulli numbers.
• Optimal Parameter Selection: The paper derives the optimal choice of the base exponent $p$ to maximize convergence rates. For odd $q = 2m+1$, the optimal choice is $p = q-1 = 2m$.
• Extension to Spectral Zeta Functions: The framework is generalized to spectral zeta functions $\zeta_L(s) = \sum \lambda_k^{-s}$ associated with eigenvalue sequences $\{\lambda_k\}$, showing that convergence depends on the interplay between the base exponent and the spectral growth rate.

4. Results and Convergence Analysis

4.1. Convergence Rates

The asymptotic error of the corrected estimator is given by:
$$B_{p,q}^n - \zeta(q) = O(n^{-\min(2p-2, \, q-1)})$$

• Standard Case ($p=2$): For any $q > 2$, the method achieves $O(n^{-2})$ convergence. This matches the rate of simple Dirichlet truncation for $q=3$ but offers a unified structural approach.
• Optimized Case ($p = q-1$): For odd $q$, choosing $p = q-1$ yields:
  $$B_{q-1,q}^n - \zeta(q) = O(n^{-(q-1)})$$
  This recovers the optimal decay rate of the truncation error while maintaining the deficiency structure.

4.2. Numerical Experiments

• Comparison: The method was tested against Dirichlet truncation and Euler–Maclaurin summation for $\zeta(3), \zeta(5), \zeta(7)$.
• Findings:
  • The corrected estimator with $p=2$ consistently achieves $O(n^{-2})$.
  • The optimized estimator ($p=q-1$) dramatically accelerates convergence, matching the theoretical $O(n^{-(q-1)})$ rate.
  • While Euler–Maclaurin remains asymptotically superior for very high orders (due to higher-order terms), the deficiency method achieves comparable high-order convergence with a simpler, implicit mechanism.
• Spectral Extension: Experiments on spectral zeta functions with eigenvalues $\lambda_k = k^\alpha$ confirmed that the method adapts effectively to different spectral growth regimes.

5. Significance and Limitations

Significance:

• Conceptual Insight: The work reframes the approximation of zeta values as a problem of "volume efficiency" under fragmentation, offering a geometric interpretation (particularly for $q=3$).
• Algorithmic Simplicity: It provides a unified, tunable scheme that avoids the symbolic complexity of deriving Euler–Maclaurin coefficients for every new exponent.
• Implicit Correction: It demonstrates that nonlinear transformations of partial sums can implicitly generate the necessary asymptotic corrections, bridging the gap between simple summation and advanced asymptotic analysis.

Limitations:

• Asymptotic Order: The method does not surpass the asymptotic order achievable by a suitably truncated Euler–Maclaurin expansion. It is not a "super-convergent" method in the strictest sense but rather an alternative structural approach.
• Closed Forms: It does not provide new closed-form expressions for odd zeta values; it remains a numerical approximation technique.

Conclusion:
The paper presents a novel, deficiency-based framework that decomposes zeta values into a nonlinear accumulation and a deficit. By optimizing the base exponent, the method achieves high-order convergence rates comparable to classical techniques but through a unified, implicit mechanism rooted in convexity and nonlinear transformations. This offers a valuable alternative perspective for both theoretical analysis and numerical computation of Dirichlet-type sums.