On the series expansion of the secondary zeta function about $s=1$ and its coefficients

1. Problem Statement

The secondary zeta function, denoted as $Z(s)$, is defined as a generalized Dirichlet series over the imaginary parts ($\gamma_n$) of the non-trivial zeros of the Riemann zeta function, assuming the Riemann Hypothesis (RH). The definition is:
$$Z(s) = \sum_{\gamma} \frac{1}{\gamma^s}$$
This series converges absolutely for $\Re(s) > 1$. The function possesses a double pole at $s=1$ and simple poles at negative odd integers.

The core problem addressed in this paper is the determination of the coefficients ($C_n$) of the Laurent series expansion of $Z(s)$ around the double pole at $s=1$. The expansion is given by:
$$Z(s) = \frac{1}{2\pi(s-1)^2} - \frac{\log(2\pi)}{2\pi(s-1)} + \sum_{n=0}^{\infty} \frac{C_n}{n!}(s-1)^n$$
While the principal part (the pole terms) was established by Chakravarty and the expansion by Ivić, a general, computable limit formula for the regular part coefficients $C_n$ for arbitrary $n \geq 0$ had not been fully derived or verified to high precision in a unified framework. Previous work by Hassani and Brent had addressed specific cases (notably $n=0$) with limited precision.

2. Methodology

The author employs a combination of analytic number theory techniques, specifically Stieltjes integration, asymptotic analysis, and summation by parts, to derive a new limit formula for $C_n$.

• Derivation of the Limit Formula:
  The proof begins by expressing the partial sums of the series derivatives using the zero-counting function $N(T)$. By utilizing the Riemann-von Mangoldt asymptotic formula $N(T) = L(T) + Q(T)$ (where $L(T)$ is the dominant term and $Q(T)$ is the error term involving the argument of the zeta function), the author transforms the discrete sum over zeros into an integral.
  Through repeated integration by parts and careful handling of boundary terms, the author isolates the divergent parts of the sum. This leads to the derivation of Theorem 1, which expresses $C_n$ as a limit:
  $$C_n = \lim_{T \to \infty} (-1)^n \left( \sum_{\gamma < T} \frac{\log^n(\gamma)}{\gamma} - \frac{1}{2\pi(n+1)(n+2)} \log^{n+1}(T) \log\left( \frac{T^{n+1}}{(2\pi)^{n+2}} \right) \right)$$
  This formula generalizes previous results (like Hassani's for $n=0$) to all $n \geq 0$.

• Convergence Improvement (BPT Theorem):
  To address the slow convergence of the direct limit formula (where the error term is $O(\frac{\log^{n+1} T}{T})$), the author applies Brent's (BPT) Theorem. By explicitly subtracting the leading error term derived from the difference between the actual zero count $N(T)$ and its asymptotic approximation $L(T)$, the error bound is improved to $O(\frac{\log^{n+1} T}{T^2})$. This yields Theorem 2, a refined formula for $C_n$ that includes a correction term involving $N(T) - L(T)$.

• Numerical Verification:
  The author utilizes a database of 2 million non-trivial zeros ($T \approx 1.13 \times 10^6$) to compute the sums directly. The results are compared against high-precision reference values (up to 50 digits) previously computed using the Arias De Reyna (ADR) algorithm.

3. Key Contributions

1. General Limit Formula (Theorem 1): The paper derives a closed-form limit expression for the Laurent coefficients $C_n$ for all $n \geq 0$. This connects the coefficients directly to the distribution of the imaginary parts of the zeros, analogous to how Stieltjes constants relate to the Riemann zeta function.
2. Convergence Acceleration (Theorem 2): The application of Brent's Theorem to the secondary zeta function context provides a significantly faster converging formula. This reduces the computational cost required to achieve high precision by effectively removing the $O(1/T)$ error term.
3. High-Precision Numerical Data: The paper provides a reference table of coefficients $C_n$ for $0 \leq n \leq 50$ computed to 50 decimal places. It also presents direct numerical verification of the derived formulas, demonstrating their accuracy.

4. Results

• Verification of $n=0$: Using the direct limit formula with 2 million zeros, the author computed $C_0 \approx 0.251637...$, matching the reference value to 5 decimal places.
• Verification of $n=1, 2$: Similar direct computations for $C_1$ and $C_2$ matched reference values to 4 and 3 decimal places, respectively, confirming the validity of the general formula.
• Impact of BPT Correction: When the Brent correction term was applied:
  • The accuracy for $C_0$ improved from 5 to 12 decimal places.
  • The accuracy for $C_1$ and $C_2$ improved to 10 decimal places.
  • The estimated error bounds dropped from $O(10^{-5})$ to $O(10^{-10})$ or better, demonstrating the efficacy of the improved formula.
• Reference Table: A comprehensive table of $C_n$ coefficients (up to $n=50$) is provided, showing the rapid growth in magnitude for higher $n$ (e.g., $C_{50} \approx -8.9 \times 10^{46}$).

5. Significance

This work bridges a gap in the analytic theory of the secondary zeta function by providing a unified method to compute its expansion coefficients.

• Theoretical Importance: It establishes a rigorous link between the asymptotic distribution of Riemann zeros and the analytic structure of $Z(s)$ near its pole, generalizing known results for the Riemann zeta function itself.
• Computational Utility: The derivation of the error-improved formula (Theorem 2) is crucial for high-precision numerical analysis. It allows researchers to compute these constants with significantly fewer zeros than previously required, making high-precision studies of the secondary zeta function more feasible.
• Foundation for Future Research: The provided reference table and the validated algorithms serve as a benchmark for future investigations into the properties of $Z(s)$, its relation to other L-functions, and potential applications in random matrix theory or spectral geometry.