Rigorous high-precision computation of the Hurwitz zeta function and its derivatives (1309.2877v1)

Abstract: We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function $\zeta(s,a)$ for $s, a \in \mathbb{C}$, along with an arbitrary number of derivatives with respect to $s$, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.

• The paper introduces a method using the Euler-Maclaurin formula to compute the Hurwitz zeta function and its derivatives with rigorous error bounds.
• It leverages advanced series expansions and parallelization to achieve high-precision results, including benchmark computations like a 303,000-digit precision of the first nontrivial zero.
• The methodology has significant implications for number theory and the study of zeta functions, with applications in computing Stieltjes constants and Keiper-Li coefficients.

Summary of "Rigorous high-precision computation of the Hurwitz zeta function and its derivatives" (1309.2877)

Introduction

The paper discusses a methodology for high-precision computation of the Hurwitz zeta function, $\zeta(s, a)$, using the Euler-Maclaurin formula. This approach allows for rigorous error bounds and the evaluation of an arbitrary number of derivatives with respect to $s$. The research presents new calculations of Stieltjes constants, Keiper-Li coefficients, and other related constants using an open-source implementation of the algorithms proposed.

Methodology

Euler-Maclaurin Formula

The numerical computation of the Hurwitz zeta function exploits the Euler-Maclaurin summation formula. This formulation provides a mechanism to evaluate $\zeta(s, a)$ for $s, a \in \mathbb{C}$ while obtaining error bounds applicable to derivatives:

$$\zeta(s,a) = \sum_{k=0}^{N-1} \frac{1}{(a+k)^s} + I + T + R$$

Here, $I$ is the integral approximation, $T$ is a correction term using the Bernoulli numbers, and $R$ is the remainder term which is minimized through appropriate selection of parameters $N$ and $M$.

Error Control

Rigorous bounds are derived for the remainder term $R$ using the periodic Bernoulli polynomials and specifically constructed integrals for precise error estimation. The paper outlines a detailed proof of these bounds, ensuring their validity across complex planes and for varying derivative orders, making them robust for high-precision applications.

Implementation Details

Algorithm Design

The implementation focuses on parallelization and fast arithmetic techniques:

• Series Expansions: Leveraging fast polynomial arithmetic (like FFT-based methods) to reduce computational overhead during power series operations.
• Parallelization: Utilizing multi-threading to improve performance during the power sum computations for large indices and precise requirements.

Computational Complexity

The asymptotic complexity for the proposed method is discussed, focusing on both theoretical bounds and practical performance, including memory use and scalability with polynomial degree $M$ and bit precision $P$.

Applications

The numerical method's efficiency is illustrated through computations of:

• Zeros of the Riemann Zeta Function: Used to approximate the first nontrivial zero to extreme precision, important in number theory and complex analysis.
• Keiper-Li Coefficients: Supporting conjectures related to the Riemann Hypothesis, further verifying these coefficients' properties.

Benchmarks

The implementation is benchmarked against other tools (mpmath in Sage and Mathematica), exhibiting superior performance particularly in computing high-precision zeros and coefficients. Notably, a 303,000-digit precision of the first nontrivial zero of the zeta function is claimed.

Conclusion

The research illustrates the practical utility of precise Hurwitz zeta function computation in mathematical and theoretical investigations. By providing a validated toolkit with rigorous error bounds, this work contributes a valuable resource for exploring the fine structure of zeta functions and their applications in pure mathematics. Extensions to this research could involve further optimization of the computational framework and exploration of alternative methods for potential gains in efficiency or applicability.