[Paper Review] Zeta functions for Riemann zeros

A. Voros|arXiv (Cornell University)|Apr 4, 2001

Analytic Number Theory Research20 references3 citations

TL;DR

This paper introduces and systematically analyzes zeta functions built from the nontrivial zeros of the Riemann zeta function, using Dirichlet series over the imaginary parts of these zeros. It establishes meromorphic continuation, computes special values and derivatives, and provides highly accurate numerical evaluations via integral approximations and functional equations, with key results including precise values for Z(σ) at critical points and a novel numerical method accelerating convergence.

ABSTRACT

A family of Zeta functions built as Dirichlet series over the Riemann zeros are shown to have meromorphic extensions in the whole complex plane, for which numerous analytical features (the polar structure, plus countably many special values) are explicitly displayed.

Motivation & Objective

• To develop a systematic, self-contained theory of zeta functions defined via Dirichlet series over the nontrivial zeros of the Riemann zeta function.
• To establish meromorphic continuation and explicit functional equations for these zeta functions, particularly for the cases v = 1/4 and v = 0.
• To compute precise numerical values and derivatives of these zeta functions across the complex plane, especially near the critical line.
• To provide a unified framework comparable to spectral zeta functions, despite the Riemann zeros' lack of a direct geometric origin.
• To offer a comprehensive reference for these generalized zeta functions, filling a gap in the literature on their explicit analytical and numerical properties.

Proposed method

• The primary zeta function is defined as 𝒵(σ,v) = ∑(τₖ² + v)⁻ᵠ for Re(σ) > 1/2, with τₖ being the imaginary parts of the nontrivial zeros.
• Meromorphic continuation to the entire complex plane is achieved via functional equations derived from the Riemann xi function and the Hurwitz zeta function.
• A modified Euler–Maclaurin formula is used, replacing the tail of the series for k > K with an integral involving the density function dN̄(T), yielding Sₖ(σ) = ∑₁^{K-1} τₖ⁻²⁰ + ½ τₖ⁻²⁰ + R̄ₖ(σ).
• The remainder term R̄ₖ(σ) is explicitly computed as (1/(2π)) τₖ¹⁻²⁰ / (2σ - 1) [log(τₖ/(2π)) + 1/(2σ - 1)], enabling fast convergence for σ > 1/2.
• For σ ≤ 1/2, continuation formulas (72)–(73) and (91) are used instead of direct summation, especially for negative or critical σ values.
• Cesaro averaging is applied to stabilize numerical fluctuations at σ ≤ 0, though it is found insufficient for deep negative σ, justifying the use of analytic continuation.

Experimental results

Research questions

• RQ1 How can zeta functions built from the Riemann zeros be systematically defined and analytically continued beyond their initial domain of convergence?
• RQ2 What are the precise numerical values and derivatives of these zeta functions at key points such as σ = 0, 1/2, 1, and -1/4?
• RQ3 How do the zeta functions 𝒵(σ,v) and 𝔻(σ,a) relate to known special functions like the Hurwitz and Lerch zeta functions?
• RQ4 What is the convergence behavior of the Dirichlet series for σ near 1/2, and how can it be accelerated numerically?
• RQ5 To what extent do these zeta functions reflect spectral-like properties, despite the absence of a geometric Laplacian?

Key findings

• The zeta function 𝒵(σ) = ∑τₖ⁻²⁰ has a finite value at σ = 1/2, with the finite part computed as approximately 0.251637.
• At σ = 0, 𝒵(0) = 0.875 exactly, matching the known value of the Riemann xi function at s = 1/2.
• The derivative of 𝒵(σ) at σ = 0 is approximately 0.8118179, and the difference between 4Z′(0) and Z(1) is less than 5×10⁻⁶, indicating strong parabolic deviation of log Ξ(s) from s(1−s).
• For σ = 1/4, 𝒵(1/4) ≈ 1.549060 and Z(1/4) ≈ 1.548829, showing close agreement between the v = 0 and v = 1/4 cases.
• The value 𝒵(−1/4) is computed as approximately 0.800805 using continuation, and 𝒵(−3/4) ≈ 1.69388, both consistent with functional equation predictions.
• The remainder term R̄ₖ(σ) effectively accelerates convergence for σ > 1/2, and stabilizes the partial sums even as σ approaches 1/2 from above.