Refined Spectral Statistics of Cubic Fields: Probing Zero Distributions via Secondary Terms

Executive Summary

The research presented in arXiv:2102.08077 provides a sophisticated analysis of the one-level density of low-lying zeros of L-functions associated with S₃-cubic fields. The central insight is the explicit derivation of secondary terms in the density formula, specifically those of the order X^-1/6. While traditional Katz-Sarnak density conjectures focus on primary asymptotic behavior as the discriminant limit X tends to infinity, this work explores the fine structure of zero distributions. This approach is promising because it provides a granular testing ground for the Generalized Riemann Hypothesis (GRH) by linking algebraic counting of fields to the spectral interpretation of zeros. The presence of a secondary term in the counting function of cubic fields—originally pioneered by Davenport and Heilbronn—propagates into the zero statistics, acting as a probe for the analytic behavior of the Riemann zeta function near the critical line.

Introduction

The study of L-functions and their zeros sits at the heart of modern analytic number theory. The Riemann Hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function ζ(s) have a real part equal to 1/2. Modern perspectives broaden this by studying families of L-functions and the statistics of their zeros near the central point s = 1/2. According to the Montgomery-Odlyzko law, these zeros are expected to follow the laws of Random Matrix Theory (RMT), matching the eigenvalues of classical compact groups.

The source paper arXiv:2102.08077 addresses the family of L-functions attached to S₃-cubic fields, denoted by F^±(X), where X is the bound on the discriminant. A distinctive feature of this family is that the counting function N^±(X) possesses a sharp secondary term of size X^5/6. This analysis demonstrates how this secondary term produces new zeta-factors in the averaged statistics, particularly involving shifts at 5/6. By refining the error terms and isolating these harmonics, the paper establishes a link between algebraic number theory and the spectral gaps of automorphic representations, offering a refined toolset for verifying the GRH in non-abelian extensions.

Mathematical Background

To analyze the one-level density, we define the family of cubic fields F^±(X) with discriminant |D_K| ≤ X. The counting result for these fields is of the form:

N^±(X) = C₁^± X + C₂^± X^5/6 + Error(X)

where C₁ and C₂ are constants determined by the Shintani zeta function. For each field K, the Dedekind zeta function factors as ζ_K(s) = ζ(s) L(s, f_K), where L(s, f_K) is the Artin L-function attached to the 2-dimensional irreducible representation of the Galois group. The one-level density D_φ(K) sums a test function φ over the scaled imaginary parts of the zeros ρ = 1/2 + iγ.

The source paper utilizes the Explicit Formula to relate these zeros to sums over primes. A crucial object is the function J^±(X), which represents the contribution of prime sums to the density. This function involves terms like (2 C₂ X^-1/6 / C₁ L) multiplied by a sum over primes weighted by p^-5e/6. The appearance of the exponent 5/6 reflects the underlying arithmetic structure of the cubic forms and creates a natural connection to the critical strip of the Riemann zeta function.

Main Technical Analysis

Spectral Properties and Secondary Zero Distribution

The analysis in arXiv:2102.08077 revolves around the decomposition of the average one-level density into a main term, a secondary term, and an error term. The derivation starts with the average of the logarithmic derivative L'/L(1/2 + r, f_K). The source shows that this average can be expressed as a series over primes involving weights f(e, s, p) and g(e, s, p), which account for the different ways a prime can ramify or split in a cubic field.

For example, the function f(e, 0, p) is defined as (e+1)/6 + (1+(-1)^e)/4 + τ_e/3 + 1/p. These local factors are essential for capturing the arithmetic fluctuations that define the secondary term. The presence of ζ'(5/6 + s) / ζ(5/6 + s) in the integral representation highlights the deep connection to the values of the Riemann zeta function away from the critical line.

Sieve-Theoretic Refinements and the X^-1/6 Term

A major technical hurdle is the treatment of the secondary term X^5/6. In the one-level density, this translates to a term in the average proportional to X^-1/6. The paper proves that this prime sum converges to a value involving the zeta function evaluated at 5/6. This is significant because the Riemann Hypothesis implies that ζ(s) has no zeros with real part greater than 1/2; therefore, ζ(5/6) is a well-defined, non-zero value. Any deviation in the distribution of cubic fields that would place a zero near s = 5/6 would directly contradict the analytic structure required for the Roberts Conjecture.

The source also provides an asymptotic for the sum over primes involving g(e, 0, p), showing it expands as 1/p^5/6+alpha + 1/p^1+2alpha. This expansion demonstrates that the secondary term is driven by the behavior of the L-function at the shifted point s = 5/6 + alpha, which is the "fingerprint" of the cubic family's secondary arithmetic.

Pole Structure and Euler Products

The paper identifies a specific constant A₄(-s, s) involving a product over primes. This constant reveals a subtle spectral gap phenomenon. The limit as s approaches 1/6 of (s - 1/6)² A₄(-s, s) is an explicit nonzero Euler product constant involving ζ(2) and ζ(5/3). This indicates a double pole at s = 1/6. Residues at such poles govern lower-order terms in the one-level density, acting as stability constraints on ζ in an off-center strip.

Novel Research Pathways

Pathway 1: Uniformity as a Constraint on Zero-Free Regions

The averaged log-derivative identity includes the term ζ'/ζ(5/6 + r). Suppose one could prove an asymptotic for the average that is uniform for r in a rectangle. Since zeros of ζ(s) off the critical line influence the size of ζ'/ζ through partial fraction expansions, a sufficiently uniform secondary term expansion would force ζ(s) to have no zeros too close to the line Re(s) = 5/6. This provides a "converse" direction where arithmetic secondary terms constrain the location of zeta zeros.

Pathway 2: Two-Level Density and Arithmetic Phase Transitions

Extending the analysis to two-level density (pair correlation) would capture correlations between pairs of zeros. This would require averaged products of two log-derivatives. The secondary term would introduce extra zeta ratios at 5/6, likely producing new "phase transitions" analogous to those seen in modular form families. This would detect whether the X^5/6 arithmetic secondary term induces a measurable deformation of pair correlation at finite conductor.

Pathway 3: Spectral Gap Amplification

The limit formula for A₄ as s approaches 1/6 reveals a concentration phenomenon. Investigating whether similar concentrations occur at other rational points related to the critical line could identify "forbidden zones" for zeros. Combining this with arithmetic geometry to understand the rational point structure may eventually constrain all zeros to lie exactly on the critical line Re(s) = 1/2.

Computational Implementation

The following Wolfram Language code demonstrates the impact of the X^-1/6 correction on the density of zeros, mirroring the mechanism used in arXiv:2102.08077 to relate prime sums to zero statistics.

(* Section: 1-Level Density with Secondary Terms *)
(* Purpose: Visualize the X^-1/6 correction from arXiv:2102.08077 *)

Module[{X, L, C1, C2, phi, mainTerm, secondaryTerm, totalDensity, plotRange},
  X = 10^7; (* Discriminant Bound *)
  L = Log[X];
  C1 = 1.0; (* Main term constant *)
  C2 = 0.45; (* Secondary term constant *)
  
  (* Test function phi(r) - Gaussian Schwartz function *)
  phi[r_] := Exp[-(r*L/(2*Pi))^2];
  
  (* Main density term (Orthogonal symmetry) *)
  mainTerm[r_] := phi[r] * (1 + Sin[r*L]/(r*L));
  
  (* Secondary term: X^-1/6 correction from the source paper *)
  secondaryTerm[r_] := -(C2/(5*C1)) * (X^(-1/6)/L) * phi[r];
  
  (* Combined density function *)
  totalDensity[r_] := mainTerm[r] + secondaryTerm[r];
  
  (* Plotting the results *)
  Plot[{mainTerm[r], totalDensity[r]}, {r, 0.01, 1.0},
    PlotRange -> All,
    PlotStyle -> {Blue, {Red, Dashed}},
    AxesLabel -> {"r", "Density"},
    PlotLegends -> {"Main Term", "With Secondary Correction"},
    PlotLabel -> "Impact of Secondary Terms on Zero Statistics"
  ]
]

Conclusions

The analysis of arXiv:2102.08077 marks a significant advancement in understanding the fine-scale distribution of L-function zeros. By isolating the X^-1/6 secondary term in the one-level density of cubic field L-functions, the authors have provided a rigorous test for the Generalized Riemann Hypothesis. These corrections introduce new global analytic factors, specifically ζ'/ζ(5/6 + s), which govern the stability of the zero distribution. The most promising avenue for further research is the numerical verification of these secondary terms, as detecting a signal of order X^-1/6 offers a unique opportunity to see the arithmetic of Shintani zeta functions reflected in the spectrum of L-functions. Ultimately, this strengthens the bridge between algebraic geometry and analytic number theory, bringing us closer to a universal understanding of the Riemann zeta function.

References

• arXiv:2102.08077: Secondary terms in the 1-level density of low-lying zeros of L-functions attached to cubic fields.
• Davenport, H., & Heilbronn, H. (1971). On the density of discriminants of cubic fields. II.
• Taniguchi, T., & Thorne, F. (2013). Secondary terms in counting functions for cubic fields.