Pair-Correlation Amplification and the Riemann Hypothesis: A Speculative Transfer from Dirichlet L-Functions

Abstract

This essay examines whether the amplification phenomenon observed by Kandhil, Languasco, and Moree—wherein pair-correlation statistics of Dirichlet L-function zeros strengthen arithmetic bounds conditional on the Generalized Riemann Hypothesis—can be transferred to the Riemann zeta function to suggest a path toward proving the Riemann Hypothesis.

Overview

This essay mines the paper Beyond the Riemann Hypothesis bounds: A pair-correlation approach to the least prime in arithmetic progression and the smallest quadratic non-residue (arXiv:2607.14515v1) for structural patterns relevant to the Riemann Hypothesis (RH). The source paper belongs to the automorphic forms and L-function family: it studies Dirichlet L-functions (GL(1) automorphic forms) and shows that combining the Generalized Riemann Hypothesis (GRH) with fine-scale vertical zero-spacing statistics (pair-correlation) yields sharper arithmetic bounds than GRH alone.

The Core Analogy

The authors define a weighted pair-correlation sum F⁺(x,T) over zeros of quadratic Dirichlet L-functions. Under GRH, they prove that this sum satisfies a strong upper bound (Theorem 1: F⁺ ≪ T log(kx)). This bound, combined with a hypothesis extending its range of uniformity (Hypothesis 1), allows them to improve classical bounds on the least quadratic non-residue n(q) to n(q) ≪ (log q)^{1+ε}.

We speculate whether an analogous bound on the Montgomery pair-correlation function for the Riemann zeta function—if provable without assuming RH—could force zeros onto the critical line. The proposed dictionary maps:

• Quadratic Dirichlet L-functions ↔ Riemann zeta function ζ(s)
• GRH assumption ↔ The Riemann Hypothesis (to be proven)
• Vertical zero spacing (pair-correlation) ↔ Montgomery's pair-correlation F(α)
• Least quadratic non-residue n(q) ↔ Error term in the prime number theorem or the Mertens function

Assessment of Strength

The correspondence rates as a formal analogy. Both sides employ sesquilinear spectral forms on zero sets, estimated via L² mean values of Dirichlet polynomials involving von Mangoldt functions. However, the analogy is not a structural isomorphism because the source domain axiomatically assumes the horizontal location of zeros (Re(s) = 1/2) to define the imaginary parts γ_j entering the sum, whereas the target domain seeks to prove this very property.

Honest Failure Mode

The critical obstruction is circularity. The source paper's pair-correlation sum F⁺ is defined as a sum over ordinates γ_j of zeros on the critical line. Transferring this definition to ζ(s) requires knowing RH to define the sum, making any proof of RH via this route circular. Furthermore, the arithmetic consequences in the source (bounds on n(q)) are extremal values modulo q, whereas the natural analog for ζ(s) (the error term in PNT) behaves differently.

The essay concludes that while the "amplification by vertical statistics" pattern is structurally resonant within the Selberg class, it does not provide a viable, non-circular bridge to prove the Riemann Hypothesis.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.