Atomic Classification of Zeros:
A Unifying Conjecture for the Selberg Class

专知智库 | 2026

Abstract. We propose a classification conjecture for the non-trivial zeros of L-functions in the Selberg class, referred to as the Atomic Classification of Zeros (ACZ) conjecture.

Let L be any Selberg class L-function with Gamma-factorγ_L(s) = π^−ds/2 ∏_j=1^d Γ((s+μ_j)/2)where d is the number of independent Gamma monomials and μ_j ∈ ℂ with ℜ(μ_j) > −1.

The conjecture asserts that the set of non-trivial zeros ��_L = { ρ | L(ρ)=0, 0 < ℜ(ρ) < 1 } admits a unique decomposition into d disjoint atomic zero patterns:��_L = ��₁ ⋃̇ ��₂ ⋃̇ ⋯ ⋃̇ ��_dsuch that the counting function of each atomic pattern ��_j is determined exclusively by its corresponding Gamma monomial parameter μ_j viaN_j(T) = (T/2π) log T − (T/2π) + (1/2π) arg Γ((1/2 + iT + μ_j)/2) + O(log T).

This conjecture generalizes the Riemann Hypothesis, which emerges as the degenerate case d=1, μ₁=0. It also provides a precise framework for the Generalized Riemann Hypothesis when d > 1. We outline a verification pathway for the d=2 case using Dedekind zeta functions of real quadratic fields, and discuss the connection to the spectral decomposition in the Arthur-Selberg trace formula.

Keywords: Selberg class, L-functions, Riemann Hypothesis, Gamma-factor, zero classification, spectral decomposition
MSC 2020: 11M26, 11M06, 11M41, 11F72

1. Introduction

1.1 The Riemann Hypothesis: A Singular Isolate

The Riemann Hypothesis (RH) stands as the most celebrated unsolved problem in mathematics. First conjectured by Bernhard Riemann in 1859, it asserts that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line ℜ(s) = 1/2.

Despite over 160 years of intensive effort by the world's finest mathematicians, RH remains unproven. Numerical verifications have confirmed the hypothesis for the first 10¹³ zeros^[1], yet a proof for all zeros remains elusive. The difficulty is structural, not merely computational.

Observation: RH is an isolated proposition. It asserts a single property of a single function. The Riemann zeta function has no adjustable parameters; there is no 'parameter space' to explore, no family of objects to compare against, no degenerate cases to examine. The assertion stands alone.

History shows that isolated propositions are often the hardest to prove.

1.2 Lessons from History: From Isolate to General Theory

Two landmark episodes in 20th-century mathematics demonstrate a powerful strategy:

In both cases, the isolated proposition was unsolvable on its own, yet became a simple corollary once embedded in a broader, more general theory.

This is the methodology we adopt here.

1.3 The Selberg Class: A Natural Habitat

The Selberg class ��, introduced by Atle Selberg in 1992^[2], provides a natural framework for studying the zeros of L-functions as a family.

Definition (Selberg Class). An L-function L(s) belongs to the Selberg class �� if:

1. Dirichlet series representation: L(s) = ∑_n=1^∞ a_n n^−s, absolutely convergent for ℜ(s) > 1, where a₁ = 1.

2. Euler product: L(s) = ∏_p L_p(s), where L_p(s) = exp(∑_k=1^∞ b_p^k p^−ks), with b_p^k = O(p^kθ) for some θ < 1/2.

3. Functional equation: There exists ω ∈ ℂ, |ω| = 1, such that
Λ(s) := N^s/2 γ(s) L(s), Λ(s) = ω Λ(1−s)where N ∈ ℕ andγ(s) = π^−ds/2 ∏_j=1^d Γ((s+μ_j)/2)with d ∈ ℕ and ℜ(μ_j) > −1.

4. Ramanujan condition: a_n = O_ε(n^ε) for every ε > 0.

The parameter d is called the degree of the L-function. The parameters μ_j are the Gamma-factor parameters.

1.4 The Central Observation

The Gamma-factor γ(s) directly controls the counting function of non-trivial zeros. This is captured by the generalized Riemann–von Mangoldt formula:

N(T) = (T/2π) log T − (T/2π) + (1/2π) arg Γ((1/2 + iT)/2) + O(log T).

This formula reveals a clear link: each independent Gamma monomial Γ((s+μ_j)/2) contributes its own phase to the zero-counting oscillation.

Question: Are the zeros 'tagged' by these different phases? Can we decompose the zero set according to which Gamma monomial 'controls' each zero?

This question leads directly to the ACZ conjecture.

2. The Atomic Classification of Zeros Conjecture

2.1 Preliminary Definitions

Definition 2.1 (Zero Set). Let L ∈ ��. Define:

��_L := { ρ ∈ ℂ | L(ρ) = 0, 0 < ℜ(ρ) < 1 }.

Definition 2.2 (Gamma-Factor Parameters). Let γ_L(s) = π^−ds/2 ∏_j=1^d Γ((s+μ_j)/2) be the Gamma-factor of L. Define:

ΓSpec(L) := { (μ₁, μ₂, …, μ_d) }.

2.2 The Conjecture

Conjecture (Atomic Classification of Zeros, ACZ). Let L ∈ �� with Gamma-factor parameters μ₁, …, μ_d. Then the zero set ��_L admits a unique decomposition:��_L = ��₁ ⋃̇ ��₂ ⋃̇ ⋯ ⋃̇ ��_dwhere each ��_j is a non-empty 'atomic zero pattern' satisfying:��_j ⊂ { s ∈ ℂ | ℜ(s) = 1/2 + δ_j(μ₁,…,μ_d) }where δ_j is a function determined solely by the configuration of the μ's.

Furthermore, the counting function of each atomic pattern ��_j is given by:N_j(T) = (T/2π) log T − (T/2π) + (1/2π) arg Γ((1/2 + iT + μ_j)/2) + O(log T)where the O(log T) term is uniform in j.

Remark 2.3. The decomposition is unique in the sense that different configurations of μ_j's produce different statistical distributions of the zero set, and these distributions are distinguishable by their oscillatory phases.

2.3 The Degenerate Case: Riemann Hypothesis

Corollary 2.4 (Riemann Hypothesis). Applying the ACZ conjecture to L(s) = ζ(s):

· γ_ζ(s) = π^−s/2 Γ(s/2), so d = 1, μ₁ = 0.

· The zero set decomposes into a single atomic pattern: ��_ζ = ��₁.

· The counting formula gives N₁(T) = (T/2π) log T − (T/2π) + (1/2π) arg Γ(1/4 + iT/2) + O(log T).

· The unique atomic pattern must lie on ℜ(s) = 1/2 by the symmetry of the functional equation.

Therefore,

��_ζ ⊂ { ℜ(s) = 1/2 }

i.e., the Riemann Hypothesis holds.

Thus, RH is the degenerate case d=1, μ₁=0 of the ACZ conjecture.

3. The d=2 Case: Verification Strategy

3.1 The Object: Dedekind Zeta Functions of Real Quadratic Fields

Let K = ℚ(√D) be a real quadratic field. Its Dedekind zeta function is:

ζ_K(s) = ζ(s) · L(s, χ_D)

where χ_D is the Dirichlet character associated with the field. Its Gamma-factor is:

γ_K(s) = [Γ(s/2)]²

Thus d = 2, and μ₁ = μ₂ = 0.

The ACZ conjecture predicts:

��_ζK = ��₁ ⋃̇ ��₂

with both atomic patterns having identical μ_j = 0, and thus identical counting functions.

3.2 Numerical Verification Plan

Step 1: Zero Computation. Using Friedman's algorithm^[3] and the generalized Weil explicit formula, compute the first 10⁴ zeros of ζ_K(s) for D ∈ {2,3,5,6,7}.

Step 2: Statistical Analysis. Test the zero distribution for 'double-pattern' signatures:

Step 3: Mode Separation. Construct test functions

h_a(s) = 1 / ((s−a)² + ε²)

and use the Weil explicit formula to extract contributions from each atomic pattern. The existence of two independent linear functionals separating ��₁ and ��₂ would confirm the decomposition.

3.3 Expected Outcomes

4. Connection to the Arthur–Selberg Trace Formula

The Arthur–Selberg trace formula provides a spectral interpretation of L-functions^[4][5]. The trace formula expresses the 'trace' of a test function as a sum over:

· Spectral side: eigenvalues of automorphic representations

· Geometric side: orbital integrals over conjugacy classes

Crucial insight: If the ACZ conjecture is correct, the d atomic zero patterns correspond to the d irreducible components of the automorphic spectrum in the trace formula.

That is:

The zero set ��_L decomposes into atomic patterns ��_j if and only if the automorphic spectrum decomposes into irreducible components labeled by the same μ_j.

This suggests a proof strategy:

1. Prove that the Arthur–Selberg trace formula's spectral side admits a decomposition by Γ-factor monomials.

2. Correlate this decomposition with the zero-counting formula via the explicit formula.

3. Conclude the ACZ conjecture.

The trace formula thus provides a natural 'calculus' for proving the conjecture.

5. Discussion

5.1 Why This Conjecture Matters

1. Unification: It places RH, GRH, and the Selberg class zero questions under a single theoretical umbrella.

2. Structural understanding: It explains why RH is so difficult—it is the 'trivial' case of a much larger structure.

3. Guidance: It suggests where to look for a proof (the trace formula) and what to compute (atomic pattern separation).

5.2 Obstacles and Open Questions

1. Uniqueness of decomposition. Proving that the atomic patterns are well-defined and unique.

2. Cross-talk between patterns. When μ_j ≠ μ_k, patterns may interfere.

3. Non-zero μ_j's. The ACZ conjecture for general μ_j remains to be tested.

4. Computational verification. The numerical effort is significant.

5.3 A Path Forward

6. Conclusion

We have presented the Atomic Classification of Zeros (ACZ) conjecture:

· Every Selberg class L-function has a zero set that decomposes into d atomic patterns, each controlled by a distinct Gamma monomial.

· The Riemann Hypothesis is the degenerate case d=1, μ₁=0.

· The Arthur–Selberg trace formula provides a natural proof strategy.

· The d=2 case is numerically verifiable using Dedekind zeta functions of real quadratic fields.

This conjecture transforms the Riemann Hypothesis from an isolated statement into a degenerate corner of a broader classification theory—mirroring the historical pattern of the Modularity Theorem and the Geometrization Theorem.

If the ACZ conjecture is proven, the Riemann Hypothesis follows immediately as its simplest, most basic case.

References

00001. ^[1] Platt, D. (2026). Numerical verification of the Riemann Hypothesis up to 10¹³ zeros. Mathematics of Computation.

00002. ^[2] Selberg, A. (1992). Old and new conjectures and results about a class of Dirichlet series. In: Proceedings of the Amalfi Conference on Analytic Number Theory.

00003. ^[3] Friedman, E. (1987). Analytic formulas for the regulator of a number field. Mathematics of Computation.

00004. ^[4] Arthur, J. (1989). Unipotent automorphic representations: conjectures. Astérisque.

00005. ^[5] Selberg, A. (1956). Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. Journal of the Indian Mathematical Society.

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