[Paper Review] Simplifications of the Keiper/Li approach to the Riemann Hypothesis

André Voros|arXiv (Cornell University)|Feb 10, 2016

Analytic Number Theory Research1 references3 citations

TL;DR

This paper introduces a new explicit sequence $Λ_n$ derived from the Keiper–Li approach to the Riemann Hypothesis, using a conformal mapping deformation to simplify the original $λ_n^\text{K}$ and $\u03bb_n^\text{L}$ sequences. The new sequence $\Lambda_n$ has a fully closed-form expression and retains sensitivity to the Riemann Hypothesis, with asymptotic behavior $\Lambda_n \sim \frac{1}{2}\log n + c$ under RH, where $c \approx -1.13033$, offering a more analytically tractable and numerically feasible alternative for testing the hypothesis.

ABSTRACT

The Keiper/Li constants $\{λ_n\}_{n=1,2,\ldots}$ are asymptotically ($n o \infty$) sensitive to the Riemann Hypothesis, but highly elusive analytically and difficult to compute numerically. We present quite explicit variant sequences that stay within the abstract Keiper--Li frame, and appear simpler to analyze and compute.

Motivation & Objective

• To simplify the Keiper–Li approach to the Riemann Hypothesis by replacing the original sequence $\lambda_n^\text{K}$ with a new, more analytically accessible variant.
• To develop a sequence $\Lambda_n$ that retains full sensitivity to the Riemann Hypothesis while avoiding the recursive complexity of prior methods.
• To provide a fully explicit, closed-form expression for $\Lambda_n$ that enables more efficient computation and analysis than the original Keiper–Li constants.
• To establish asymptotic behavior of $\Lambda_n$ under the Riemann Hypothesis, showing $\Lambda_n \sim \frac{1}{2}\log n + c$ with $c \approx -1.13033$.
• To explore alternative basepoints, such as $x = \frac{1}{2}$, and derive a centered variant $\Lambda_n^0(\tilde{w})$, though with less favorable algebraic simplicity.

Proposed method

• Deforms the Keiper–Li framework by replacing the conformal mapping $M(z) = (1 - z)^{-1}$ with a new analytic function to generate a new sequence $\Lambda_n$.
• Derives a closed-form expression for $\Lambda_n$ using residue calculus and a transformed variable $r$, leading to the formula $\Lambda_n = \sum_{m=1}^n \frac{2}{(r_m+1)^2} \frac{\prod_{k=0}^n (r_m + r_k)}{\prod_{k \neq m} (r_m - r_k)} \log 2\xi(2m)$.
• Uses the functional equation of $\xi(x)$ and analytic continuation to ensure the new sequence remains sensitive to the location of nontrivial zeros.
• Applies the saddle-point method and asymptotic analysis to derive the large-$n$ behavior of $\Lambda_n$ under both RH and its failure.
• Introduces a centered variant $\Lambda_n^0(\tilde{w})$ using a different conformal mapping centered at $x = \frac{1}{2}$, though with less favorable algebraic structure.
• Validates the new sequence numerically and compares it to known results, showing agreement with the asymptotic prediction $\Lambda_n \sim \frac{1}{2}\log n + c$ under RH.

Experimental results

Research questions

• RQ1 Can the Keiper–Li approach to the Riemann Hypothesis be simplified through a reparameterization of the generating function's conformal mapping?
• RQ2 Does a fully explicit, closed-form sequence exist that preserves the Riemann Hypothesis sensitivity of the original $\lambda_n^\text{K}$ and $\lambda_n^\text{L}$ sequences?
• RQ3 What is the asymptotic behavior of the new sequence $\Lambda_n$ under the assumption of the Riemann Hypothesis?
• RQ4 Can the new sequence $\Lambda_n$ be computed more efficiently than the original Keiper–Li constants due to its closed-form structure?
• RQ5 How does the performance and sensitivity of the new sequence compare to the original under both RH and its failure, especially in the large-$n$ regime?

Key findings

• The new sequence $\Lambda_n$ is explicitly given by a closed-form formula involving residues and rational functions of $r_m = \sqrt{1 + (4m - 1)^2 / \tilde{w}}$, enabling direct computation without recursion.
• Under the Riemann Hypothesis, $\Lambda_n \sim \frac{1}{2}\log n + c$ with $c = \frac{1}{2}(\gamma - \log 2\pi - 1) \approx -1.130330700754$, matching the asymptotic behavior of the original $\lambda_n^\text{L}$.
• The sequence $\Lambda_n$ remains sensitive to the Riemann Hypothesis: if RH is false, $\Lambda_n$ grows faster than any $o(n^\varepsilon)$, with $\log|\Delta_\rho \Lambda_n| \sim (\rho - \frac{1}{2})\log n$ for zeros off the critical line.
• Numerical evaluation shows that $\Lambda_n$ behaves consistently with the theoretical asymptotics, with the same leading-order logarithmic growth and constant term as the original Keiper–Li sequence.
• A centered variant $\Lambda_n^0(\tilde{w})$ is derived, with asymptotic behavior $\Lambda_n^0(\tilde{w}) \sim \sqrt{\tilde{w}}(\log n + C)$, $C \approx -0.78375711$, but with less favorable algebraic structure for analysis.
• The new sequence avoids the $\sim \frac{1}{4}$ decimal place precision loss per step seen in recursive evaluations of $\lambda_n^\text{L}$, significantly improving numerical feasibility.