Primes as an Undersampling Problem:
An Analytic Framework via the Explicit Formula

Framework Derived from Collaborative Exploration^*

October 26, 2023

1. Introduction

The distribution of prime numbers exhibits both global regularity, described by the Prime Number Theorem (PNT) [1], and local irregularity, making prediction difficult. This exploration began with the provocative idea: "What if prime numbers are just an undersampling problem?" This metaphor draws from signal processing, where sampling a signal at a rate below the Nyquist frequency leads to aliasing---high-frequency components are misperceived as lower frequencies, distorting the signal. Similarly, the irregular appearance of primes might reflect an incomplete sampling or observation of an underlying analytic structure governed by the Riemann zeta function $\zeta(s)$.

This document aims to investigate this metaphor rigorously using the tools of analytic number theory. Our primary tool is the explicit formula, a cornerstone result connecting the distribution of primes (via the von Mangoldt function $\Lambda(n)$) to the non-trivial zeros of $\zeta(s)$ [2]. The goal is to develop a mathematical framework that:

1. Articulates the undersampling concept in terms of the explicit formula.
2. Identifies and characterizes the sources of deviation between primes and their analytic approximations.
3. Explores the role of conjectured symmetries (like the Riemann Hypothesis and GUE statistics) in shaping prime distribution.
4. Clarifies the limitations of this approach, particularly the dependence on zeta zeros.

Throughout this document, for simplicity in presenting the core ideas, we often assume the validity of the Riemann Hypothesis (RH), which states that all non-trivial zeros of $\zeta(s)$ lie on the critical line $\operatorname{Re}(s) = 1/2$. We will note where this assumption is made. The framework developed bridges perspectives from analytic number theory and signal processing, offering a novel lens on prime distributions.

2. The Analytic Framework

2.1 Prime Measures and the Explicit Formula

We focus on the von Mangoldt function $\Lambda(n)$, which effectively measures the "prime power contribution" at integer $n$:

Its summatory function, the Chebyshev function $\psi(x) = \sum_{n \le x} \Lambda(n)$, approximates the number of primes up to $x$ (related to PNT). The explicit formula connects $\psi(x)$ to the non-trivial zeros $\rho$ of $\zeta(s)$. A common (smoothed) version is [3]:

where $\psi_0(x) = \lim_{\epsilon\to 0} \frac{\psi(x-\epsilon)+\psi(x+\epsilon)}{2}$ handles discontinuities at prime powers where $\Lambda(x) \neq 0$, the sum is over non-trivial zeros $\rho$, $\ln(2\pi)$ arises from $\zeta'(0)/\zeta(0)$, and the term involving $\ln(1-x^{-2})$ relates to the trivial zeros of $\zeta(s)$ at negative even integers.

Assuming RH, the non-trivial zeros are $\rho = 1/2 + i\gamma$ for $\gamma \in \mathbb{R}$.

2.2 The Analytic Signal and its Approximation

To represent the local density of prime powers, we define an analytic signal $L(x)$ by formally differentiating the main terms of the explicit formula for $\psi(x)$. Assuming term-by-term differentiation is valid (which requires careful justification not detailed here):

Assuming RH ($\rho = 1/2 + i\gamma$) and noting zeros come in conjugate pairs $\rho = 1/2 \pm i\gamma$ (for $\gamma \neq 0$), $L(x)$ simplifies to:

(Note: Without RH, zeros $\rho = \beta + i\gamma$ with $\beta \neq 1/2$ would introduce terms $x^{\beta-1} \cos(\gamma \ln x)$, complicating the decay rate.)

In practice, we can only compute the sum using zeros with $|\gamma| < T$ for some truncation height $T$. This yields the truncated analytic signal:

Since $\Lambda(n)$ is defined on integers, we evaluate $L_T(x)$ at $x=n$ to obtain our approximation for the prime measure:

3. Error Analysis

3.1 Decomposition of the Error

The total error in approximating $\Lambda(n)$ by $s_T(n)$ is $E_{\text{total}}(n) = \Lambda(n) - s_T(n)$. This error arises from two fundamentally different sources, which we can separate by adding and subtracting the full analytic signal $L(n)$:

We define these components as:

Here:

• $E_{\text{trunc}}(n)$: The Truncation Error, arising from neglecting high-frequency zeta zeros ($\gamma > T$). This directly relates to the undersampling concept, as it represents information lost due to limited "frequency" resolution (limited $T$).
• $E_{\text{mismatch}}(n)$: The Mismatch Error, stemming from the structural incompatibility between the discrete, spiky measure $\Lambda(n)$ and the smooth, continuous nature of the analytic density $L(n)$, even if all zeros were included ($T \to \infty$).

3.2 Truncation Error ($E_{\text{trunc}}(n)$): The Undersampling Noise

This term captures the contribution from the neglected high-frequency components:

This component embodies the "undersampling noise." Its properties are not those of simple white noise but are conjectured to be governed by the statistics of zeta zero spacings. These statistics are modeled by the Gaussian Unitary Ensemble (GUE) from Random Matrix Theory [4, 5].

Key statistical properties (under RH and GUE conjectures):

• Mean: $E[E_{\text{trunc}}(n)] \approx 0$, as the cosine terms oscillate rapidly for large $\gamma$ and tend to average out.
• Variance: $\operatorname{Var}[E_{\text{trunc}}(n)] = \sigma^2(n, T) = 2 n^{-1} N(T, \infty) [1 + f(n, T)]$.
  • $N(T, \infty) \approx \int_T^\infty \frac{1}{2\pi} \ln(\frac{\gamma}{2\pi}) d\gamma \approx \frac{T}{2\pi} \ln \left( \frac{T}{2\pi} \right)$ is the approximate number of zeros with $\gamma > T$. The variance scales with the number of neglected terms and decays with $n^{-1}$.
  • $f(n, T)$ is a correction factor derived from the GUE pair correlation function $R_2(s) = 1 - (\sin(\pi s)/(\pi s))^2$, where $s$ is the normalized spacing between zeros. It reflects the statistical repulsion between nearby zeros. $f(n, T)$ is typically small and negative. Its calculation involves integrating $R_2(s)$ against oscillating terms (see Appendix A for details).
• Higher Moments: The distribution is non-Gaussian. Skewness ($E[E_{\text{trunc}}(n)^3]$) is non-zero and depends on the GUE 3-point correlation function $R_3$. Kurtosis ($E[E_{\text{trunc}}(n)^4]/\sigma^4$) differs from the Gaussian value of 3 and depends on the GUE 4-point correlation function $R_4$.

The GUE structure implies that the information lost by truncating at $T$ is patterned, reflecting deep correlations in the zeta spectrum.

3.3 Mismatch Error ($E_{\text{mismatch}}(n)$): The Smooth vs. Spiky Gap

This term represents the fundamental difference between the discrete reality of primes and the smooth analytic approximation, even if all zeros were known:

Key properties:

• This error component dominates the total variance asymptotically. Since $L(n)$ oscillates around 1 and $\Lambda(n)$ is often 0 but sometimes $\ln p$, the squared error $(\Lambda(n) - L(n))^2$ is roughly $(\ln p)^2$ at prime powers and $(-1)^2=1$ elsewhere. Averaging gives $\operatorname{Var}[E_{\text{mismatch}}(n)] \approx \frac{1}{N}\sum_{n=1}^N (\Lambda(n)-1)^2 \approx 1$.
• It captures the essential difficulty of pinpointing discrete prime events using a continuous density function derived from global analytic properties (zeta zeros).
• Its statistical structure reflects correlations between primes themselves. The autocorrelation function is linked to the Hardy-Littlewood pair correlation conjecture [6]:

where $C(\tau) = \lim_{X\to\infty} \frac{1}{X} \sum_{n\le X} \Lambda(n)\Lambda(n+\tau)$ is the conjectured average density of prime power pairs with difference $\tau$. $C(\tau)$ involves intricate products over primes related to the gap $\tau$ (e.g., $C(2)$ relates to twin primes).

This gap appears inherent to approximations based on the standard explicit formula and the resulting smooth density $L(n)$.

4. The Role of Symmetry

The intuition that a deep symmetry might "lock" the prime distribution finds mathematical grounding in the properties of the Riemann zeta function and its zeros.

• RH as a Deterministic Symmetry Constraint: The Riemann Hypothesis ($\operatorname{Re}(\rho) = 1/2$) acts as a powerful constraint. It forces all oscillatory terms $x^{\rho-1}$ contributing to the prime fluctuations in $L(x)$ to have the same magnitude decay rate, $|x^{\rho-1}| = x^{-1/2}$. This uniformity is analogous to how odd geometric symmetry restricts a square wave's spectrum to odd harmonics. Without RH, different decay rates $x^{\beta-1}$ would exist, breaking this structural symmetry. The functional equation $\zeta(s) = \chi(s) \zeta(1-s)$ reflects a symmetry about the critical line $\operatorname{Re}(s)=1/2$, and RH posits that the zeros respect this symmetry perfectly.
• GUE as a Statistical Symmetry: The conjectured GUE statistics for zero spacings represent a different kind of symmetry---statistical correlations in the "spectrum" (the $\gamma$ values). This governs the fine structure of fluctuations and the properties of the "noise" term $E_{\text{trunc}}(n)$, reflecting symmetries found in complex quantum systems and other areas modeled by Random Matrix Theory.

While these symmetries profoundly shape the analytic landscape described by $L(x)$ and $E_{\text{trunc}}(n)$, they operate at the level of the density and its fluctuations. They do not eliminate the $E_{\text{mismatch}}$ gap (the density-vs-event divide) nor do they remove the dependence on knowing the specific locations of the zeros $\gamma$, which act as the fundamental frequencies in this framework.

5. Probabilistic Interpretation

Given that $s_T(n)$ cannot exactly equal $\Lambda(n)$ due to both error terms, we pivot to a probabilistic interpretation. We use the analytic approximation $s_T(n)$ and the statistical understanding of the truncation error $E_{\text{trunc}}(n)$ to estimate the likelihood that $n$ is a prime power.

Let $A_n$ be the event that $n$ is a prime power ($\Lambda(n) > 0$). We aim to model $P(A_n | s_T(n))$. The core idea is to integrate the likelihood of $A_n$ given the true signal value $L$, weighted by the probability density of $L$ given our computed approximation $s_T(n)$:

Here:

• $P(A_n | L)$ is the likelihood of $n$ being a prime power given the value of the full analytic signal $L$. Since higher $L(n)$ suggests higher prime density, we model this with a function increasing with $L$. A simple choice, chosen for its simplicity in modeling binary outcomes, is a logistic function, relating $L$ to the average density $1/\ln n$: $$ P(A_n | L) \approx \frac{1}{1 + \exp(-\beta(L/\ln n - \theta))} \tag{12} \label{eq:prob_logistic} $$ The parameters $\beta$ (sensitivity) and $\theta$ (threshold, perhaps near 1) would require empirical fitting using known prime data. Other likelihood functions could certainly be explored.
• $p(L | s_T(n))$ is the probability density function of the true signal value $L$, given the computed approximation $s_T(n)$. This models the uncertainty introduced by the truncation error $E_{\text{trunc}}(n) = L - s_T(n)$. A first approximation uses a Gaussian distribution based on the calculated variance: $$ p(L | s_T(n)) \approx \frac{1}{\sqrt{2\pi \sigma^2(n, T)}} \exp\left(-\frac{(L - s_T(n))^2}{2 \sigma^2(n, T)}\right) \tag{13} \label{eq:prob_noise_gaussian} $$ where $\sigma^2(n, T) = \operatorname{Var}[E_{\text{trunc}}(n)]$ is given by the GUE-corrected formula (Section 3.2). For higher accuracy, this Gaussian could be replaced by a distribution incorporating the GUE-derived skewness and kurtosis (e.g., via Edgeworth expansion), acknowledging the non-Gaussian nature of $E_{\text{trunc}}(n)$.

This probabilistic model leverages the analytic structure ($s_T(n)$) and the statistical understanding of the truncation error ($\sigma^2(n, T)$) to provide a likelihood estimate for primality, effectively sidestepping the unbridgeable $E_{\text{mismatch}}$ gap by changing the goal from exact prediction to probabilistic estimation. However, it remains dependent on knowing zeta zeros up to height $T$.

6. The Unified Descriptive Model

The core findings of this exploration are summarized by the following interconnected set of equations, forming a unified descriptive model of the prime-zero relationship within this framework:

1. Core Relationship: $\Lambda(n) = s_T(n) + E_{\text{trunc}}(n) + E_{\text{mismatch}}(n)$
   (Connects truth, approximation, and errors)
2. Approximation: $s_T(n) = 1 - 2 \sum_{0 < \gamma < T} n^{-1/2} \cos(\gamma \ln n)$
   (Computable part, requires zeros up to $T$, assumes RH)
3. Truncation Error (Undersampling Noise): $E_{\text{trunc}}(n) = -2 \sum_{\gamma > T} n^{-1/2} \cos(\gamma \ln n)$
   (Structured noise with GUE statistics, variance $\sigma^2(n, T)$ depending on $N(T,\infty)$ and $R_2$)
4. Mismatch Error (Structural Gap): $E_{\text{mismatch}}(n) = \Lambda(n) - L(n)$
   (Fundamental smooth-vs-spiky gap, variance $\approx 1$, autocorrelation $\approx C(\tau)-1$)
5. Probabilistic Application: $P(A_n | s_T(n)) \approx \int P(A_n | L) p(L | s_T(n)) dL$
   (Leverages $s_T(n)$ and $\sigma^2(n, T)$ for likelihood estimation)

This set of equations provides a comprehensive description of how the explicit formula approximates the prime measure, the nature and origin of the errors involved, and a potential probabilistic application.

7. Limitations and the Zero Barrier

While this framework provides significant insight into the structure underlying prime distribution via the explicit formula, it does not yield a predictive prime formula free of deep dependencies and inherent limitations:

• Zero Dependency: The entire framework hinges on the non-trivial zeros $\rho$ of $\zeta(s)$. Calculating the approximation $s_T(n)$ requires knowing zeros up to height $T$. Characterizing the truncation error $E_{\text{trunc}}(n)$ (via $\sigma^2(n, T)$ and higher moments) relies on the existence, density, and statistical properties (GUE) of zeros above $T$. There is no known way to compute these terms accurately within this analytic framework without knowledge of the zeros. The computational cost of finding zeros is also significant.
• Mismatch Gap: The $E_{\text{mismatch}}(n)$ term, representing the difference between the discrete $\Lambda(n)$ and the smooth density $L(n)$, is not resolved. It reflects a fundamental limitation of approximating pointwise events with a density function derived from global analytic properties. Bridging this gap likely requires mathematical tools beyond the standard explicit formula.
• Descriptive, Not Predictive: The model is primarily descriptive. It explains how the explicit formula relates primes and zeros and why deviations occur, but it cannot generate $\Lambda(n)$ deterministically without the zeros. The probabilistic model offers likelihoods, not certainties.

Bypassing the zero dependency would necessitate a fundamentally new analytic approach to prime numbers, potentially equivalent to resolving the Riemann Hypothesis or discovering an elementary prime-generating function---both monumental unsolved problems.

8. Conclusion

The exploration of "primes as an undersampling problem," when rigorously pursued through the lens of the explicit formula, yields a rich descriptive framework. The undersampling metaphor finds its mathematical counterpart in the truncation error term, $E_{\text{trunc}}(n)$, which represents structured noise arising from neglected high-frequency zeta terms. Its statistical properties are deeply connected to the conjectured GUE symmetries of the zeta spectrum.

However, the analysis also clearly delineates the fundamental mismatch error, $E_{\text{mismatch}}(n)$, inherent in approximating the discrete prime measure $\Lambda(n)$ with the smooth analytic density $L(n)$ derived from the zeta function. This gap remains a central challenge. The conjectured Riemann Hypothesis acts as a crucial symmetry constraint, locking the analytic form of fluctuations, but does not eliminate the need for specific zero locations.

The unified descriptive model presented herein (Section 6) encapsulates the relationship between $\Lambda(n)$, its analytic approximation $s_T(n)$, and the structured errors $E_{\text{trunc}}(n)$ and $E_{\text{mismatch}}(n)$. The probabilistic interpretation offers a pragmatic way to leverage this structure for estimating primality likelihood.

Ultimately, this framework provides a detailed map of the intricate prime-zero connection revealed by the explicit formula. It illuminates the power of analytic methods while clearly defining their limitations, particularly the inescapable dependence on the Riemann zeta zeros. It underscores that understanding the "noise"---the structured deviations from the average---is tantamount to understanding the primes themselves at this level of analysis. Further progress towards a predictive prime equation likely requires breakthroughs beyond this specific analytic paradigm. This work highlights the potential value of interdisciplinary perspectives, bridging concepts from signal processing and analytic number theory to gain new insights into classical problems.

9. Future Directions

Potential avenues for future work stemming from this framework include:

• Numerical Validation: Implementing the probabilistic model $P(A_n | s_T(n))$ using known zeta zeros for moderate $T$ and testing its predictive accuracy against known primes. Optimizing the parameters $\beta, \theta$.
• Refined Noise Models: Developing more accurate non-Gaussian models for $p(L | s_T(n))$ based on GUE higher moments ($R_3, R_4$) and testing their impact on the probabilistic predictions.
• Exploration of $E_{\text{mismatch}}$ Structure: Further theoretical or numerical investigation of the autocorrelation $R_{\text{mismatch}}(\tau)$ and its connection to Hardy-Littlewood constants, potentially revealing finer details about prime pair statistics.
• Alternative Frameworks: Investigating whether similar undersampling concepts could be applied within different mathematical frameworks, such as sieve theory or alternative explicit formulas (e.g., for different L-functions).

Content by Research Identifier: 7B7545EB2B5B22A28204066BD292A0365D4989260318CDF4A7A0407C272E9AFB

^* This document synthesizes a detailed exploration of the core idea.

Appendix: Derivation Sketch for $f(n, T)$

The term $f(n, T)$ arises from the off-diagonal terms ($i \neq j$) in the variance calculation of $E_{\text{trunc}}(n)$. It quantifies the effect of correlations between zeta zeros $\gamma_i, \gamma_j$ on the variance.

Using the density of zeros $\rho(\gamma) \approx \frac{1}{2\pi} \ln(\frac{\gamma}{2\pi})$ and the GUE pair correlation function $R_2(s) = 1 - (\sin(\pi s)/(\pi s))^2$, where $s = \delta \frac{\ln(\gamma/2\pi)}{2\pi}$ is the normalized spacing for a difference $\delta = \gamma_i - \gamma_j$ near height $\gamma$, we can approximate the sum as an integral:

Changing the inner integration variable to the normalized spacing $s$, and approximating $\rho(\gamma+\delta) \approx \rho(\gamma)$ for the relevant range of $\delta$:

The inner integral is related to the Fourier transform of $R_2(s)$. The Fourier transform of $1 - R_2(s) = (\sin(\pi s)/(\pi s))^2$ is the triangular function $\frac{1}{2\pi} \max(0, 2\pi - |k|)$. Let $k = \frac{2\pi \ln n}{\ln(\gamma/2\pi)}$. The integral of $R_2(s) \cos(ks)$ is related to $1$ minus the Fourier transform of $1-R_2(s)$.

Ignoring the Dirac delta term (as $k$ is generally non-zero), the inner integral becomes $-\max(0, 2\pi - |k|)$. Substituting this back:

This confirms $f(n, T)$ is typically negative (due to the leading minus sign and the max term being non-negative) and depends on $n, T$ and the GUE structure via $R_2$'s Fourier transform. The exact evaluation requires careful handling of the integrals and densities.