A Proposed Unified Framework for Detecting GRH Violations via L-function Signatures

Abstract: We propose a unified framework to detect violations of the Generalized Riemann Hypothesis (GRH) by analyzing perturbations in arithmetic sums associated with L-functions. Building on the Aliasing Signature (AS) framework [6, 7], we introduce a general equation modeling the effect of hypothetical off-critical-line zeros, characterized by a structure factor $\SL(q, a; \rho)$ derived from L-function twists. We test this framework numerically on L-functions $L(E, s)$ associated with two distinct elliptic curves: the non-CM curve $E_1: y^2 = x^3 + 7$ and the CM curve $E_2: y^2 = x^3 - x$. Simulating the effect of a hypothetical off-critical-line zero $\rho_0 = 0.55 + 14.13i$, we explore the resulting signatures. Applying the simplifying "generic assumption" for $\SL$, correlation analysis reveals a positive correlation signature ($+1.0$) for both curves modulo $q=4$ and $q=5$. However, acknowledging the potential failure of this assumption for the CM curve $E_2$, we also simulate hypothetical alternative scenarios where non-trivial twist interactions alter $\SL$. For $q=4$, assuming $S_L(4,3) = -S_L(4,1)$ results in an anti-correlation signature ($-1.0$). For $q=5$, assuming the quadratic character contributes leads to zero correlation between certain residue class pairs. This stark contrast demonstrates the framework's sensitivity to $\SL$ and underscores the critical need to determine this factor accurately based on the L-function's arithmetic structure (especially CM). While promising, the framework requires significant theoretical development and validation. Our results suggest a potential signal processing strategy for identifying GRH violations.

Keywords: Generalized Riemann Hypothesis, L-functions, Elliptic Curves, Complex Multiplication, Explicit Formula, Aliasing Signature, Signal Processing, Computational Number Theory
MSC2020: 11M26, 11G05, 11G15, 11N13, 11Y35, 94A12

1. Introduction

L-functions stand as central objects in modern number theory, weaving together arithmetic data from diverse sources such as number fields, elliptic curves, modular forms, and Galois representations. The Generalized Riemann Hypothesis (GRH) conjectures that all non-trivial zeros of any suitably defined L-function $L(s)$ lie on a critical line, typically $\re(s) = 1/2$. This conjecture, if true, implies deep structural regularity in the distribution of primes and related arithmetic objects [1]. Conversely, a violation of GRH—the existence of a zero $\rho_0 = \sigma_0 + it_0$ with $\sigma_0 > 1/2$—would dramatically alter our understanding of these distributions.

While GRH remains unproven, significant efforts have explored its consequences and sought counterexamples [2, 3, 4, 5]. This paper introduces a different perspective: a framework designed to predict and detect the specific, structured signature that an individual hypothetical off-critical-line zero would imprint onto associated arithmetic counting functions when analyzed modulo $q$.

Building upon the Aliasing Signature (AS) concept [6, 7], we propose a unified framework applicable across various L-function families. This framework models the dominant error term from a GRH violation using a general equation characterized by a structure factor $\SL(q, a; \rho)$. This factor encapsulates how the L-function's arithmetic properties, particularly its behavior under twisting by characters $\pmod q$, determine the correlation patterns across residue classes $a \pmod q$.

We conduct numerical case studies using L-functions associated with $E_1: y^2 = x^3 + 7$ (non-CM) and $E_2: y^2 = x^3 - x$ (CM). Simulating the effect of $\rho_0 = 0.55 + 14.13i$, we first apply a simplifying "generic assumption" for $\SL$. Recognizing its limitations, particularly for the CM curve $E_2$, we then simulate hypothetical alternative signatures for $E_2$ for $q=4$ and $q=5$, assuming non-trivial interactions with twists alter $\SL$. We analyze the results using correlation analysis and illustrative Fast Fourier Transform (FFT) techniques.

Our key finding is the framework's sensitivity to $\SL$. Under the generic assumption, both curves yield identical $+1.0$ correlation signatures for $q=4, 5$. However, the alternative simulations for $E_2$ produce drastically different signatures ($-1.0$ correlation for $q=4$, zero correlation between certain pairs for $q=5$). This underscores the necessity of rigorously deriving $\SL$ for specific L-function families. Despite the preliminary nature of this work, the results suggest the AS framework, combined with signal processing, could provide a useful diagnostic tool for investigating GRH.

2. Background: L-functions, GRH, and the Explicit Formula

An L-function $L(s) = \sum \lambda_L(n) n^{-s}$ typically has analytic continuation, a functional equation relating $L(s)$ to $L(k-s)$, and an Euler product. GRH posits non-trivial zeros lie on the critical line $\re(s) = k/2$. For elliptic curves, $k=2$, critical line $\re(s)=1$, but we often use an effective critical line $\re(s)=1/2$ for GRH analysis [1].

The explicit formula [1] connects sums over coefficients $\lambda_L(n)$ to sums over zeros $\rho$: \[ \label{eq:explicit_schematic} \tag{1} \sum_{n \le x} \lambda_L(n) \Lambda_{\text{eff}}(n) = M_L(x) - \sum_{\rho} \frac{x^\rho}{\rho} + \text{other terms}. \] GRH failure via $\rho_0 = \sigma_0 + it_0$ ($\sigma_0 > 1/2$) leads to dominant $\Order(x^{\sigma_0})$ fluctuations.

Definition 2.1 (Modular Arithmetic Sum). For an L-function $L(s)$ with coefficients $\lambda_L(n)$, the modular arithmetic sum is: \[ \sum_{\substack{n \leq x \\ n \equiv a \pmod{q}}} \lambda_L(n) \Lambda_{\text{eff}}(n). \] For elliptic curves $E_1, E_2$, we denote these sums as $\psiLEone(x; q, a), \psiLEtwo(x; q, a)$.

3. The Unified Aliasing Signature Framework

We model the contribution of a GRH-violating pair $(\rho_0, \rho_1 = 1-\overline{\rho_0})$ to the modular arithmetic sum.

Framework 3.1 (Proposed Unified Aliasing Signature). Assume GRH fails via $\rho_0 = \sigma_0 + it_0$ ($\sigma_0 > 1/2$). The dominant perturbation is: \[ \label{eq:AS_unified} \tag{2} \ASterm(x; q, a; \rho_0) \approx - \left( \frac{x^{\rho_0}}{\rho_0} + \epsilon \frac{x^{\rho_1}}{\rho_1} \right) \SL(q, a; \rho_0) \] where $\epsilon \approx 1$ and $\SL(q, a; \rho_0)$ is the Structure Factor: \[ \label{eq:structure_factor} \tag{3} \SL(q, a; \rho_0) = \frac{1}{\phi(q)} \sum_{\psi \pmod{q}} \overline{\psi(a)} w_L(\psi, \rho_0). \] $w_L(\psi, \rho_0)$ reflects if $\rho_0$ is a zero of $L(s, \psi)$. The "generic assumption" posits $w_L(\psi_0)=1, w_L(\psi \neq \psi_0)=0$. Total sum: $\psiL'(x; q, a) \approx \psiL(x; q, a)_{\text{GRH}} + \re(\ASterm(x; q, a; \rho_0))$.

Hypothesis 3.2 (Properties of the Aliasing Signature). $\ASterm$ has amplitude $\Order(x^{\sigma_0})$, oscillates at $f_0 = t_0 \ln(10) / (2\pi)$ vs $\log_{10} x$, and modular structure depends on $\SL(q, a; \rho_0)$.

Remark 3.3 (Caveats on $w_L$). The generic assumption for $w_L$ may fail, especially for structured L-functions (e.g., CM). Rigorous determination of $w_L$ and $\SL$ is crucial.

4. Case Studies: Elliptic Curve L-functions

We simulate $\rho_0 = 0.55 + 14.13i$ for $E_1$ (non-CM) and $E_2$ (CM) for $q=4, 5$. We compare the generic assumption against hypothetical alternatives for $E_2$. Effective critical line $\sigma_c=1/2$. We use `numPoints = 1000` for better resolution.

4.1 Case Study 1: Non-CM Curve $E_1: y^2 = x^3 + 7$

4.1.1 Setup ($E_1$)

• Curve: $E_1: y^2 = x^3 + 7$ (LMFDB 14.a4, Cond. 14, non-CM).
• Sum: $\psiLEone(x; q, a)$.
• Generic Assumption: Plausible starting point.

4.1.2 Aliasing Signature Prediction ($E_1$, Generic Assumption)

Generic assumption ($w_{L_1}(\psi_0)=1, w_{L_1}(\psi \neq \psi_0)=0$) yields:

• $q=4$: $\SL(4, 1) = 1/2$, $\SL(4, 3) = 1/2 \implies +1.0$ Correlation.
• $q=5$: $\SL(5, a) = 1/4$ for $a=1, 2, 3, 4 \implies +1.0$ Correlation.

4.1.3 Simulation Results ($E_1$)

Simulating $\psiLEone'(x; q, a)$ with simple baselines and AS terms based on the generic assumption.

Results confirm the +1.0 correlation signature predicted by the generic assumption for $E_1$.

4.2 Case Study 2: CM Curve $E_2: y^2 = x^3 - x$

4.2.1 Setup ($E_2$)

• Curve: $E_2: y^2 = x^3 - x$ (LMFDB 32.a2, Cond. 32, CM by $\mathbb{Z}[i]$).
• L-function: $L(E_2, s)$ is a Hecke L-function. $a_p(E_2) = 0$ if $p=2$ or $p \equiv 3 \pmod 4$.
• Sum: $\psiLEtwo(x; q, a)$. Note $\psiLEtwo(x; 4, 3)_{\text{GRH}} \approx 0$.
• Generic Assumption Plausibility: Unlikely due to CM structure.
• Baseline Models: Note: Baseline models for $E_2$ used below are illustrative, employing $\Order(x^{1/2})$ noise (except for $a=3 \pmod 4$ which is correctly zero). Future work should calibrate these using actual $a_n(E_2)$ coefficients from LMFDB [11] for greater realism.

4.2.2 Aliasing Signature Prediction ($E_2$)

We explore three scenarios:

1. Scenario A (Generic Assumption, $q=4, 5$): Assume $w_{L_2}(\psi_0)=1, w_{L_2}(\psi \neq \psi_0)=0$.
   • $q=4$: $\SL(4, 1) = 1/2$, $\SL(4, 3) = 1/2 \implies +1.0$ Correlation.
   • $q=5$: $\SL(5, a) = 1/4$ for $a=1, 2, 3, 4 \implies +1.0$ Correlation.
2. Scenario B (Alternative, $q=4$): Assume CM structure implies $S_L(4,1)=1/2$ and $S_L(4,3)=-1/2$. Predicts **-1.0 correlation** for $q=4$.
3. Scenario C (Alternative, $q=5$): Assume CM structure implies the quadratic character $\chi_{5,2}$ contributes with weight $w=1$ alongside $w_0=1$. This yields $\SL(5, 1) = 1/2$, $\SL(5, 2) = 0$, $\SL(5, 3) = 0$, $\SL(5, 4) = 1/2$. Predicts zero AS term for $a=2, 3$, leading to zero correlation between pairs involving $a=2$ or $a=3$, and +1.0 correlation between $a=1, 4$.

4.2.3 Simulation Results ($E_2$)

Simulating $\psiLEtwo'(x; q, a)$ for the different scenarios.

Scenario A (Generic Assumption, $q=4$): Uses $S_L(4,1)=1/2, S_L(4,3)=1/2$.

Scenario B (Alternative, $q=4$): Uses $S_L(4,1)=1/2, S_L(4,3)=-1/2$.

Scenario A (Generic Assumption, $q=5$): Uses $S_L(5,a)=1/4$.

Scenario C (Alternative, $q=5$): Uses $S_L(5,1)=S_L(5,4)=1/2, S_L(5,2)=S_L(5,3)=0$.

The simulations for $E_2$ clearly show how different assumptions about $\SL$ lead to distinct correlation signatures (+1.0, -1.0, or 0 depending on the scenario and pair).

4.3 Fourier Analysis

The AS term introduced by $\rho_0 = 0.55 + 14.13i$ oscillates at frequency $f_0 = t_0 \ln(10) / (2\pi) \approx 5.178$ cycles per unit of $\log_{10} x$. Figure 7 shows an illustrative power spectrum based on this frequency.

5. Discussion

The numerical simulations across the non-CM ($E_1$) and CM ($E_2$) elliptic curves provide critical insights into the proposed Unified Aliasing Signature Framework and the pivotal role of the structure factor $\SL$.

Sensitivity to $\SL$: The framework's core prediction mechanism, Eq. (\ref{eq:AS_unified}), depends directly on $\SL(q, a; \rho_0)$. Our simulations vividly demonstrate this sensitivity:

• For $E_1$ (non-CM), the generic assumption yields +1.0 correlations for $q=4, 5$ (Figs 1-2, Table 1).
• For $E_2$ (CM), the generic assumption *also* yields +1.0 correlations (Figs 3, 5, Table 2).
• However, simulating plausible alternative scenarios for $E_2$ reflecting potential CM structure yields drastically different results: a -1.0 correlation for $q=4$ (Fig 4, Table 2) and zero correlation for pairs involving $a=2, 3$ for $q=5$ (Fig 6, Table 2).

The Challenge of the Generic Assumption (especially for CM): While the generic assumption ($w_L(\psi \neq \psi_0)=0$) provides a useful baseline, its validity must be assessed case-by-case. It is significantly suspect for CM curves like $E_2$. Our alternative simulations illustrate possible outcomes if $w_{L_2}(\psi, \rho_0) \neq 0$. Applying this framework rigorously requires theoretical work to determine the actual weights $w_L$ for the specific L-function family.

Detection Potential & Practical Challenges: The simulations confirm correlation analysis and frequency detection (Fig. 7) as viable strategies. Due to the dominance of the AS term ($\Order(x^{\sigma_0})$) over baseline noise ($\Order(x^{1/2})$) for large $x$, correlations approach $\pm 1.0$ or 0 strongly in these idealized simulations. Practical challenges remain: accurate computation of sums, real-world noise, sampling issues, and the theoretical determination of $\SL$.

6. Conclusion and Future Directions

We proposed a unified AS framework using the structure factor $\SL(q, a; \rho_0)$ to predict signatures of hypothetical GRH violations. Case studies on non-CM ($E_1$) and CM ($E_2$) elliptic curves illustrate its mechanics and sensitivity.

Simulations show the framework predicts distinct signatures based on $\SL$. While the generic assumption yields +1.0 correlations for both $E_1$ and $E_2$, plausible alternative scenarios for the CM curve $E_2$ produce markedly different signatures (-1.0 or mixed correlations). This highlights the critical need for rigorous determination of $\SL$, moving beyond the generic assumption, especially for structured L-functions.

This framework offers a novel diagnostic tool, providing specific predictions to guide computational searches. Its preliminary nature necessitates significant future work:

1. Theoretical Refinement of $\SL$: Rigorously derive $w_L(\psi, \rho_0)$ and $\SL$ for various L-function families.
2. Broader Testing & Baseline Calibration: Apply the framework with validated $\SL$ values and realistic baseline models calibrated with actual coefficient data.
3. Computational Implementation & Real Data Analysis: Develop efficient code and apply to actual L-function data.
4. Signal Processing Enhancements: Improve techniques for noise reduction and signature extraction (e.g., using actual FFTs on real data).

By linking potential GRH violations to concrete signatures sensitive to arithmetic structure via $\SL$, this framework aims to advance the search for counterexamples and deepen our understanding of L-function zeros.

Appendix A: Derivation Sketch of the Structure Factor

(Sketch remains unchanged)

Using character orthogonality for $\gcd(a, q)=1$: \[ \sum_{\substack{n \leq x \\ n \equiv a \pmod{q}}} \lambda_L(n) \Lambda_{\text{eff}}(n) \approx \frac{1}{\phi(q)} \sum_{\psi \pmod{q}} \overline{\psi(a)} \left( \sum_{n \leq x} \lambda_L(n) \psi(n) \Lambda_{\text{eff}}(n) \right) \] The inner sum relates to zeros of the twisted L-function $L(s, \psi)$. The contribution of a specific zero $\rho_0$ involves summing over twists $\psi$ where $\rho_0$ is a zero, weighted by $w_L(\psi, \rho_0)$: \[ \text{Contribution}(\rho_0) = - \frac{x^{\rho_0}}{\rho_0} \left( \frac{1}{\phi(q)} \sum_{\psi \pmod{q}} \overline{\psi(a)} w_L(\psi, \rho_0) \right) = - \frac{x^{\rho_0}}{\rho_0} \SL(q, a; \rho_0) \] Including the symmetric zero $\rho_1$ gives Eq. (\ref{eq:AS_unified}).

References

• [1] Iwaniec, H. and Kowalski, E. Analytic Number Theory. AMS, 2004.
• [2] Montgomery, H. L. "The pair correlation of zeros..." Proc. Sympos. Pure Math., XXIV, 1973.
• [3] Odlyzko, A. M. "On the distribution of spacings..." Math. Comp. 48 (1987).
• [4] Platt, D. J. "Isolating the zeros..." Math. Comp. 86 (2017).
• [5] Guth, L., & Maynard, J. "Zero density for L-functions." arXiv:2403.04947 (2024).
• [6] Author Alias. "The Aliasing Signature Framework..." In preparation (2025).
• [7] Author Alias. "Aliasing Signatures in Dirichlet L-functions." In preparation (2024).
• [8] Silverman, J. H. Advanced Topics in the Arithmetic of Elliptic Curves. Springer, 1994.
• [9] Cremona, J. E. Algorithms for Modular Elliptic Curves. 2nd ed. CUP, 1997.
• [10] The LMFDB Collaboration. Elliptic curve 14.a4 ($E_1$). LMFDB.
• [11] The LMFDB Collaboration. Elliptic curve 32.a2 ($E_2$). LMFDB.
• [12] The LMFDB Collaboration. The L-functions and Modular Forms Database. LMFDB.

Note: References [6, 7] are forthcoming. The AS framework is presented here with a self-contained derivation sketch in Appendix A.