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

Artur Kawalec

Abstract

The secondary zeta function is defined as a generalized zeta series over the imaginary parts of non-trivial zeros assuming (RH). This function admits Laurent series expansion at the double pole at $s=1$. In this article, we derive a new formula for the expansion coefficients of the regular part, which is similar to the Stieltjes constants formula for the Riemann zeta function. We also numerically verify and compute the new formula to high precision for several test cases. Lastly, we also apply the Brent's (BPT) Theorem for improving convergence of the main formula.

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

Abstract

The secondary zeta function is defined as a generalized zeta series over the imaginary parts of non-trivial zeros assuming (RH). This function admits Laurent series expansion at the double pole at . In this article, we derive a new formula for the expansion coefficients of the regular part, which is similar to the Stieltjes constants formula for the Riemann zeta function. We also numerically verify and compute the new formula to high precision for several test cases. Lastly, we also apply the Brent's (BPT) Theorem for improving convergence of the main formula.
Paper Structure (4 sections, 2 theorems, 47 equations, 1 table)

This paper contains 4 sections, 2 theorems, 47 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1
  3. Numerical Computation
  4. Application of (BPT) for improving error bound in (3)

Key Result

Theorem 1

for $n\geq 0$.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2