Cohomological Density of Primes over Spec (Z) : A Sheaf-Theoretic and Geometric Framework

1. Introduction

In analytic number theory, the distribution of prime numbers has been a central object of study for centuries. The prime counting function $\pi \left(x\right)$, which counts the number of primes less than or equal to x, encodes deep arithmetic structure and continues to serve as a touchstone for both classical and modern inquiries. From Gauss’s empirical observations to the formal proof of the Prime Number Theorem (PNT), significant progress has been made in understanding the asymptotic behavior of primes through analytic techniques involving complex analysis and the Riemann zeta function.

However, recent developments in algebraic geometry and homological algebra invite a reconsideration of classical number-theoretic objects from a geometric and cohomological perspective. In particular, the spectrum of the integers, $Spec\left(\mathbb{Z}\right)$, endowed with the Zariski topology, offers a topological and categorical framework in which primes naturally arise as closed points. This opens the possibility of interpreting prime distributions as geometric density over schemes, and even more precisely, as supports of coherent or constructible sheaves.

This paper aims to construct a cohomological reinterpretation of prime density by leveraging the tools of sheaf theory, étale topology, and derived functors. We define and study specific sheaves whose support is concentrated on prime ideals of $\mathbb{Z}$, and analyze their behavior through stalks, global sections, and cohomology groups $H^i$. A novel notion of cohomological density is introduced, which generalizes traditional asymptotic density and offers a more flexible algebraic and geometric formulation.

Furthermore, we explore how this framework can be extended from the base scheme $Spec\left(\mathbb{Z}\right)$ to more general polynomial rings such as $\mathbb{Z}\left[x\right]$, with an emphasis on prime values taken by special polynomials such as $f\left(n\right)=n^2+1$. The aim is to show how the distribution of prime outputs from such polynomials can also be interpreted via sheaf-theoretic methods, thus unifying analytic, geometric, and topological approaches to the prime distribution problem.

1.1. Structure of the Paper

• In Section 2, we begin with a classical review of the prime counting function $\pi \left(x\right)$ and the analytic techniques used to study its growth.
• Section 3 develops the topological framework of $Spec\left(\mathbb{Z}\right)$ and the role of open sets $D\left(f\right)$ in characterizing the localization of primes.
• In Section 4, we introduce the étale topology and investigate how primes behave under more refined sheaf-theoretic operations.
• Section 5 formalizes the notion of cohomological density and relates it to the existence and vanishing of global sections and cohomology groups.
• Section 6 generalizes the results to polynomial settings and demonstrates their applicability through the case study of $n^2+1$.
• Finally, Section 7 summarizes our contributions and outlines future directions in the interplay between number theory and modern algebraic geometry.

2. Classical Prime Counting and Analytic Methods

2.1. Definition and Historical Origins of the Prime Counting Function $\pi \left(x\right)$

The prime counting function, denoted by $\pi \left(x\right)$, is defined as follows:

$$\pi \left(x\right)=\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{primes}p\le x,x\in \mathbb{R},x\ge 2.$$

This deceptively simple definition encapsulates one of the most profound and historically significant functions in analytic number theory. The function $\pi \left(x\right)$ captures the fundamental question regarding the distribution of prime numbers among the integers.

Historically, the study of prime numbers dates back to antiquity, with significant contributions from Greek mathematicians such as Euclid, who proved the infinitude of primes. However, the explicit quantitative study of prime distribution began notably with Leonhard Euler in the 18th century. Euler initiated systematic studies of prime distributions through his seminal product formula linking primes to the Riemann zeta function:

$$\zeta \left(s\right)=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{n^s}}=\prod _p{\displaystyle \frac{1}{1-p^{-s}}},Re\left(s\right)>1.$$

Euler’s work laid the groundwork for understanding primes through analytic methods and connected prime distribution intimately with the theory of special functions and infinite products.

The prime counting function itself gained considerable attention through the work of Carl Friedrich Gauss at the end of the 18th century. Around 1792, Gauss empirically observed the approximate asymptotic behavior of primes, suggesting the following approximation for large x:

$$\pi \left(x\right)\approx Li\left(x\right)={\int }_2^x{\displaystyle \frac{dt}{logt}},$$

where $Li\left(x\right)$ is the logarithmic integral function. Although Gauss never provided a formal proof, his extensive numerical tables supported this remarkable conjecture, now known to be closely related to the Prime Number Theorem (PNT).

Throughout the 19th century, various mathematicians including Legendre, Chebyshev, and ultimately Hadamard and de la Vallée-Poussin significantly refined the understanding of $\pi \left(x\right)$’s asymptotic behavior, culminating in the rigorous proof of the Prime Number Theorem in 1896. This theorem asserts that:

$$\pi \left(x\right)\sim {\displaystyle \frac{x}{logx}},\mathrm{as}x\to \infty ,$$

confirming Gauss’s original intuition regarding prime distribution.

This historical trajectory—from Euler’s initial analytic insights, Gauss’s empirical observations, to the rigorous establishment of PNT—highlights the prime counting function’s central role in analytic number theory. Understanding $\pi \left(x\right)$ not only involves deep analytic tools but also offers intriguing connections to algebraic and geometric structures, which will be explored further in subsequent sections of this paper.

2.2. Numerical Computation, Graphical Representation, and Accuracy Analysis of the Approximation $Li\left(x\right)$

The prime counting function $\pi \left(x\right)$ is of central importance not only theoretically but also numerically, as precise calculations offer valuable insight into prime distributions. Historically, various numerical methods have been employed to approximate $\pi \left(x\right)$, with the logarithmic integral function $Li\left(x\right)$ serving as the primary asymptotic estimator:

$$Li\left(x\right)={\int }_2^x{\displaystyle \frac{dt}{logt}}.$$

Gauss initially conjectured, based on extensive numerical experiments, that $Li\left(x\right)$ closely approximates $\pi \left(x\right)$ with remarkable accuracy for large values of x. Empirical data indeed shows this approximation holds extraordinarily well, even though precise deviations occur, a phenomenon deeply studied in analytic number theory. The deviation between $\pi \left(x\right)$ and $Li\left(x\right)$ is quantitatively expressed as:

$$\pi \left(x\right)-Li\left(x\right).$$

Extensive numerical analyses by mathematicians such as Riemann, Littlewood, and Hardy established that this difference changes sign infinitely often as $x\to \infty$.

2.2.1. Numerical Computation and Accuracy

The following table provides a numerical comparison of $\pi \left(x\right)$, $x/logx$, and $Li\left(x\right)$, along with their errors for various magnitudes of x:

Table 1. Numerical comparison of $\pi \left(x\right)$, $x/logx$, and $Li\left(x\right)$.

x	$\mathit{\pi }\left(\mathit{x}\right)$	$\mathit{\pi }\left(\mathit{x}\right)-\frac{\mathit{x}}{log\mathit{x}}$	$Li\left(\mathit{x}\right)-\mathit{\pi }\left(\mathit{x}\right)$	Error (%)	$\frac{\mathit{x}}{\mathit{\pi }\left(\mathit{x}\right)}$
10	4	0	2	8.22	2.500
${10}^2$	25	3	5	14.06	4.000
${10}^3$	168	23	10	14.85	5.952
${10}^4$	1229	143	17	12.37	8.137
${10}^5$	9592	906	38	9.91	10.425
${10}^6$	78498	6116	130	8.11	12.739
${10}^7$	664579	44158	339	6.87	15.047
${10}^8$	5761455	332774	754	5.94	17.357
${10}^9$	50847534	2592592	1701	5.23	19.667
${10}^{10}$	455052511	20758029	3104	4.66	21.975

Table 1. Numerical comparison of $\pi \left(x\right)$, $x/logx$, and $Li\left(x\right)$.

x	$\mathit{\pi }\left(\mathit{x}\right)$	$\mathit{\pi }\left(\mathit{x}\right)-\frac{\mathit{x}}{log\mathit{x}}$	$Li\left(\mathit{x}\right)-\mathit{\pi }\left(\mathit{x}\right)$	Error (%)	$\frac{\mathit{x}}{\mathit{\pi }\left(\mathit{x}\right)}$
10	4	0	2	8.22	2.500
${10}^2$	25	3	5	14.06	4.000
${10}^3$	168	23	10	14.85	5.952
${10}^4$	1229	143	17	12.37	8.137
${10}^5$	9592	906	38	9.91	10.425
${10}^6$	78498	6116	130	8.11	12.739
${10}^7$	664579	44158	339	6.87	15.047
${10}^8$	5761455	332774	754	5.94	17.357
${10}^9$	50847534	2592592	1701	5.23	19.667
${10}^{10}$	455052511	20758029	3104	4.66	21.975

These numerical results clearly demonstrate the high accuracy of $Li\left(x\right)$ as an approximation for $\pi \left(x\right)$, despite minor deviations that decrease proportionally with increasing values of x.

2.2.2. Quantitative Error Analysis

The error percentage between $Li\left(x\right)$ and $\pi \left(x\right)$ diminishes as x grows. The following asymptotic behavior describes this relation clearly:

$$\underset{x\to \infty }{lim}{\displaystyle \frac{\pi \left(x\right)-Li\left(x\right)}{\pi \left(x\right)}}=0.$$

This indicates that $Li\left(x\right)$ remains an exceptional approximation of $\pi \left(x\right)$, with numerical and graphical analyses validating the theoretical results developed through analytic number theory.

2.3. Structural Interpretation via $\theta \left(x\right)$ and $\psi \left(x\right)$

The prime counting function $\pi \left(x\right)$ itself offers a fundamental measure of prime distribution, yet its structural behavior is more explicitly captured through certain closely related auxiliary functions. Specifically, two functions, Chebyshev’s theta function $\theta \left(x\right)$ and Chebyshev’s psi function $\psi \left(x\right)$, provide deeper analytic and structural insights into the distribution of primes.

2.3.1. Definition of Chebyshev’s Functions

Chebyshev’s theta function, denoted $\theta \left(x\right)$, is defined as:

$$\theta \left(x\right)=\sum _{p\le x}logp,$$

where the summation runs over all prime numbers $p\le x$. Similarly, Chebyshev’s psi function, $\psi \left(x\right)$, extends this definition to include prime powers:

$$\psi \left(x\right)=\sum _{n\le x}\Lambda \left(n\right),$$

where $\Lambda \left(n\right)$ is the von Mangoldt function, defined as:

$$\Lambda \left(n\right)=\left\{\begin{array}{cc}logp\hfill & \mathrm{if}n=p^k,p\mathrm{prime}\mathrm{and}k\ge 1,\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

2.3.2. Structural Relationship Between $\pi \left(x\right)$, $\theta \left(x\right)$, and $\psi \left(x\right)$

These functions relate to $\pi \left(x\right)$ through explicit and implicit structural identities. The most direct connections can be seen via the Möbius inversion formula and partial summation methods. For instance, partial summation yields:

$$\pi \left(x\right)={\displaystyle \frac{\theta \left(x\right)}{logx}}+{\int }_2^x{\displaystyle \frac{\theta \left(t\right)}{t{(logt)}^2}}dt,$$

and similarly:

$$\psi \left(x\right)=\theta \left(x\right)+O\left(\sqrt{x}{log}^{2}x\right).$$

These integral and summation representations highlight how $\pi \left(x\right)$, $\theta \left(x\right)$, and $\psi \left(x\right)$ mutually encapsulate prime distributions, though each emphasizes different structural aspects of primes.

2.3.3. Analytic and Structural Significance

The structural importance of $\theta \left(x\right)$ and $\psi \left(x\right)$ becomes particularly evident through their relationship with the Riemann zeta function $\zeta \left(s\right)$. It is known from analytic number theory that:

$$\psi \left(x\right)=x-\sum _{\rho }{\displaystyle \frac{x^{\rho }}{\rho }}-{\displaystyle \frac{{\zeta }^{\prime }\left(0\right)}{\zeta \left(0\right)}}-{\displaystyle \frac{1}{2}}log\left(1-x^{-2}\right),$$

where the summation runs over the nontrivial zeros $\rho$ of the zeta function $\zeta \left(s\right)$. This explicit formula directly connects the distribution of prime numbers with complex analytic structures, especially the critical zeros of the Riemann zeta function.

2.3.4. Interpretation and Structural Insights

The functions $\theta \left(x\right)$ and $\psi \left(x\right)$ offer valuable insight into the oscillatory behavior observed in prime distributions. They smooth out the irregularities that naturally arise when counting primes explicitly (as with $\pi \left(x\right)$), thus providing more robust tools for theoretical exploration and proving asymptotic results.

Furthermore, the connection of these functions with zeros of the Riemann zeta function demonstrates how prime distribution intricately intertwines with deep properties in complex analysis and number theory, suggesting profound underlying structures and relationships yet to be fully understood.

In summary, while the prime counting function $\pi \left(x\right)$ provides a straightforward numeric measure of prime density, the structural richness of prime distribution reveals itself most vividly through $\theta \left(x\right)$ and $\psi \left(x\right)$. This deeper understanding sets the stage for subsequent geometric and cohomological interpretations explored in later chapters.

2.4. Historical Overview and Statement of the Prime Number Theorem (PNT)

The Prime Number Theorem (PNT), a cornerstone result of analytic number theory, describes the asymptotic behavior of the prime counting function $\pi \left(x\right)$. Formally, the Prime Number Theorem states:

$$\pi \left(x\right)\sim {\displaystyle \frac{x}{logx}}\mathrm{as}x\to \infty .$$

This result means that the ratio between $\pi \left(x\right)$ and the function $x/logx$ approaches 1 as x grows without bound, symbolically represented as:

$$\underset{x\to \infty }{lim}{\displaystyle \frac{\pi \left(x\right)}{x/logx}}=1.$$

2.4.1. Historical Development

The intuition behind this profound asymptotic statement dates back to Carl Friedrich Gauss, who in the late 18th century observed through numerical evidence the remarkable approximation of prime distribution by the logarithmic integral function $Li\left(x\right)$. However, formal proofs remained elusive until the late 19th century.

Adrien-Marie Legendre first provided explicit conjectures and rough approximations, refining Gauss’s observations. In the mid-19th century, Pafnuty Chebyshev made significant strides by proving bounds that confirmed the correct order of magnitude for prime distribution:

$$c_1{\displaystyle \frac{x}{logx}}\le \pi \left(x\right)\le c_2{\displaystyle \frac{x}{logx}},$$

for constants $c_1,c_2$ close to 1. Although Chebyshev’s results did not establish the exact limit, they significantly narrowed the mathematical gap, suggesting the imminent proximity to the Prime Number Theorem.

2.4.2. Proof of the PNT

The precise asymptotic equivalence was rigorously proven independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin in 1896. Their groundbreaking proofs fundamentally utilized complex analysis, specifically the properties of the Riemann zeta function $\zeta \left(s\right)$. Both mathematicians leveraged the analytic continuation and zeros of the zeta function to deduce the exact asymptotic distribution of primes, marking one of the earliest and most profound intersections between analytic number theory and complex analysis.

In particular, their arguments hinged upon demonstrating that the zeta function has no zeros on the critical vertical line $Re\left(s\right)=1$, a fundamental insight that allowed them to control the error terms precisely enough to establish the asymptotic equivalence of the prime counting function and $x/logx$.

2.4.3. Mathematical Significance and Subsequent Impact

The Prime Number Theorem not only resolved a fundamental conjecture in mathematics but also established analytic number theory as a powerful field capable of answering deep arithmetic questions through complex-analytic methods. Its proof methods and subsequent generalizations laid the groundwork for even more profound conjectures, such as the Riemann Hypothesis, which to this day remains unproven and deeply influences the landscape of mathematics.

This historical development clearly illustrates both the significance and the analytical complexity of the Prime Number Theorem. It sets a stage for further exploration into deeper analytic, geometric, and algebraic methods of understanding prime distribution, as pursued throughout subsequent sections of this paper.

2.5. Complex-Analytic Interpretation of the Prime Number Theorem via the Zeta Function

The Prime Number Theorem owes much of its mathematical depth and significance to its intimate connection with the Riemann zeta function:

$$\zeta \left(s\right)=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{n^s}}=\prod _p{\left(1-{\displaystyle \frac{1}{p^s}}\right)}^{-1},\mathrm{for}Re\left(s\right)>1.$$

The zeta function acts as an analytic encoding of the distribution of prime numbers and plays a critical role in the proof of the PNT through its behavior on the complex s-plane.

2.5.1. Analytic Continuation and Functional Equation

Originally defined only for $Re\left(s\right)>1$, the zeta function can be analytically continued to the entire complex plane, except for a simple pole at $s=1$. This continuation allows the application of contour integration and complex-analytic tools in regions where the original series does not converge. Furthermore, the Riemann zeta function satisfies a functional equation:

$$\zeta \left(s\right)=2^s{\pi }^{s-1}sin\left({\displaystyle \frac{\pi s}{2}}\right)\Gamma (1-s)\zeta (1-s),$$

which reflects the function across the critical line $Re\left(s\right)={\displaystyle \frac{1}{2}}$ and deeply influences its symmetry.

2.5.2. Poles and Zeros of the Zeta Function

The zeta function has a simple pole at $s=1$ with residue 1, which directly connects to the divergence of the harmonic series and plays a central role in the asymptotic formula for $\pi \left(x\right)$. It has nontrivial zeros in the critical strip $0<Re\left(s\right)<1$, and the Riemann Hypothesis conjectures that all such zeros lie on the critical line $Re\left(s\right)={\displaystyle \frac{1}{2}}$. Hadamard and de la Vallée-Poussin’s proof of the PNT relied on showing that:

$$\zeta \left(s\right)\ne 0\mathrm{for}\mathrm{all}s\in \mathbb{C}\mathrm{such}\mathrm{that}Re\left(s\right)=1.$$

This result eliminates certain types of singularities in the Perron integral formula for $\pi \left(x\right)$ and ensures precise control over the error term in its asymptotic expansion.

2.5.3. Explicit Formula and the Role of Zeros

A profound insight arises from the explicit formula, which expresses $\psi \left(x\right)$ in terms of the zeros $\rho$ of $\zeta \left(s\right)$:

$$\psi \left(x\right)=x-\sum _{\rho }{\displaystyle \frac{x^{\rho }}{\rho }}+(\mathrm{lower}\mathrm{order}\mathrm{terms}).$$

Here, the sum runs over nontrivial zeros $\rho =\beta +i\gamma$, revealing that the oscillations in prime distribution are encoded by the imaginary parts of the zeros.

This formula illuminates how the PNT corresponds to the dominant term x, and how the nonvanishing of $\zeta \left(s\right)$ on the line $Re\left(s\right)=1$ guarantees that the sum over $x^{\rho }/\rho$ does not contribute leading order terms that could violate the $x/logx$ behavior of $\pi \left(x\right)$.

2.6. Limitations of Analytic Approaches and the Need for Geometric Transition

2.6.1. Intrinsic Limitations of Analytic Methods

The standard proof of the PNT and its refinements are heavily reliant on tools from classical complex analysis: analytic continuation, functional equations, and zero-free regions of $\zeta \left(s\right)$. However, these techniques are fundamentally asymptotic in nature:

• They describe the global density of primes but offer limited control over local distribution (e.g., exact patterns, gaps).
• The error terms in the asymptotic expansions (e.g., in the explicit formula for $\psi \left(x\right)$) are deeply connected to the yet unresolved Riemann Hypothesis, showing a fundamental barrier to precision.
• Analytic approaches are generally non-constructive: while they establish the existence of certain distributions or density, they often do not provide algebraic or geometric mechanisms explaining why the primes are distributed as such.

These limitations motivate a search for a more structurally descriptive approach.

2.6.2. Motivation for a Geometric Perspective

Algebraic geometry, particularly through the lens of scheme theory, provides a categorical and topological reinterpretation of number-theoretic objects:

• The spectrum $Spec\left(\mathbb{Z}\right)$ encodes primes as geometric points.
• Open sets $D\left(f\right)$ define natural neighborhoods of primes.
• The Zariski topology introduces a coarse but algebraically meaningful structure.

In this setting, primes can be interpreted not just as isolated integers but as geometric loci within a topological framework, paving the way for sheaf-theoretic and cohomological methods to study their distribution.

3. Topological Framework of $Spec\left(\mathbb{Z}\right)$

3.1. Primes as Closed Points

In the scheme $Spec\left(\mathbb{Z}\right)$, prime ideals $\left(p\right)$ correspond to closed points, and the generic point $\left(0\right)$ corresponds to the fraction field $\mathbb{Q}$. The Zariski topology defines open sets as $D\left(f\right)=\left\{\right(p)\mid f\notin (p\left)\right\}$.

3.2. Role of Open Sets $D\left(f\right)$

Open sets $D\left(f\right)$ localize the arithmetic information around primes not dividing f. This coarse topology limits the resolution of local arithmetic structures, necessitating a finer topology like the étale topology.

3.3. Zariski Density of Primes

The set of all primes $\mathcal{P}$ is Zariski-dense in $Spec\left(\mathbb{Z}\right)$, as every non-empty open set $D\left(f\right)$ contains infinitely many primes unless $f=0$.

4. Étale Topology and Sheaf-Theoretic Analysis

4.1. Introduction to Étale Topology

The étale topology refines the Zariski topology by allowing morphisms that are flat and unramified, capturing finer arithmetic data via Galois actions and local extensions.

4.2. Étale Sheaves and Their Stalks

An étale sheaf F on $Spec\left(\mathbb{Z}\right)$ assigns data to étale morphisms $U\to Spec\left(\mathbb{Z}\right)$. The stalk $F_p$ at a prime $\left(p\right)$ reflects the local Galois module structure.

4.3. Galois Action and Local Data

The stalk $F_p$ carries an action of the local Galois group $G_{{\mathbb{Q}}_p}=Gal({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$, encoding arithmetic information like ramification and inertia.

4.4. Analysis of Étale Sheaf Behavior at a Fixed Prime p

4.4.1. Étale Neighborhoods of $\left(p\right)$

Étale neighborhoods of $\left(p\right)$ involve morphisms like $Spec(\mathbb{Z}\left[x\right]/(x^2-p))\to Spec\left(\mathbb{Z}\right)$.

4.4.2. Étale Sheaf Restriction: ${\left.F\right|}_{\left(p\right)}$

The restriction reflects whether F is constant, splits, or has monodromy.

4.4.3. Ramification, Inertia, and Fiber Behavior

If the inertia subgroup $I_p$ acts trivially on $F_p$, the sheaf is unramified; otherwise, it is ramified.

4.5. Ramification and Its Effect on Regularity of Étale Sheaves

4.5.1. Definition of Regularity in Sheaf-Theoretic Terms

A sheaf F is étale-regular at $\left(p\right)$ if there exists an étale neighborhood $U\to Spec\left(\mathbb{Z}\right)$ such that ${\left.F\right|}_U$ is locally constant and unramified, equivalent to $F_p$ being invariant under $I_p$.

4.5.2. Ramification Obstructs Regularity

Let F be an étale sheaf over the arithmetic scheme $X=Spec\left(\mathbb{Z}\right)$. The notion of regularity at a point $\left(p\right)\in X$ is closely tied to the sheaf’s behavior under étale localizations and the action of the absolute Galois group $Gal({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$.

Definition .

Let F be an étale sheaf on $X=Spec\left(\mathbb{Z}\right)$. We say F isétale-regularat a closed point $\left(p\right)$ if there exists an étale neighborhood $U\to X$ such that ${\left.F\right|}_U$ is locally constant and unramified at p. Equivalently, the stalk $F_p$ is invariant under the inertia subgroup $I_p\subset Gal({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$.

Theorem .

Let F be a constructible étale sheaf on $X=Spec\left(\mathbb{Z}\right)$. If F is ramified at a closed point $\left(p\right)$, then:

(1)

The inertia group $I_p$ acts non-trivially on the stalk $F_p$.

(2)

F cannot be locally trivialized in any étale neighborhood of $\left(p\right)$.

(3)

The canonical map $\Gamma (U,F)\to F_p$ is not surjective for any étale neighborhood U containing $\left(p\right)$.

Proof .

(1)

Since F is ramified at $\left(p\right)$, the stalk $F_p$ is a module over the local Galois group $G_{{\mathbb{Q}}_p}=Gal({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$. By definition of ramification, the inertia subgroup $I_p$ acts non-trivially on $F_p$. Consider an étale morphism $U=Spec\left(\mathbb{Z}\left[\sqrt{p}\right]\right)\to X$. The pullback ${\left.F\right|}_U$ has a stalk at $\left(p\right)$ that inherits the $I_p$-action, which is non-trivial by assumption, preventing $F_p$ from being invariant.

(2)

Local triviality requires that ${\left.F\right|}_U$ is locally constant for some étale neighborhood $U\to X$. If such a U exists, the stalk $F_p$ must be constant under the étale pullback, implying a trivial $I_p$-action. Since $I_p$ acts non-trivially, this is a contradiction, so F cannot be locally trivialized.

(3)

Consider an étale neighborhood $U\to X$ containing $\left(p\right)$. The map $\Gamma (U,F)\to F_p$ sends global sections to the stalk at $\left(p\right)$. Since F is ramified, $F_p$ carries a non-trivial $I_p$-action, and sections in $\Gamma (U,F)$ must be invariant under the Galois group of U. By the non-trivial action, no section in $\Gamma (U,F)$ can map surjectively to $F_p$, as the image is restricted to the $I_p$-invariant subspace, which is strictly smaller than $F_p$.

□

Corollary .

Ramification at $\left(p\right)$ introduces singularities in the arithmetic geometry of F, detectable in $H_{\text{é}\mathrm{t}}^1(X,F)$, reflecting local torsors or failure of Galois descent.

Example .

Let F be the étale sheaf associated with the Galois module ${\mu }_n$ over $X=Spec\left(\mathbb{Z}\right[1/n\left]\right)$. Then F is ramified at primes dividing n, and F fails to be regular at those primes. This is reflected in the non-vanishing of $H_{\text{é}\mathrm{t}}^1(Spec\left(\mathbb{Z}[1/n]\right),{\mu }_n)\cong \mathbb{Z}/n\mathbb{Z}$.

4.5.3. Consequences for Arithmetic Geometry

The presence of ramification in an étale sheaf F over $X=Spec\left(\mathbb{Z}\right)$ has profound implications for both local and global structures in arithmetic geometry. This section explores the categorical and cohomological consequences of non-regularity caused by ramification.

(1)

Obstruction to Descent: Let F be an étale sheaf ramified at a prime $\left(p\right)\subset Spec\left(\mathbb{Z}\right)$. Then:

• F fails to descend along finite étale covers, especially those trivializing at p.
• The Galois action on the stalk $F_p$ is non-trivial under the inertia subgroup $I_p$, obstructing the existence of a global trivialization.

(2)

Modified Galois Representations: The stalk $F_p$ may be viewed as a representation of the local Galois group $G_{{\mathbb{Q}}_p}=Gal({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$. Ramification implies:

$${\rho }_{F,p}:G_{{\mathbb{Q}}_p}\to Aut\left(F_p\right)$$

is not trivial on $I_p$, hence ${\rho }_{F,p}$ encodes ramified information that modifies the global Galois representation ${\rho }_F:G_{\mathbb{Q}}\to Aut\left(F\right)$.

(3)

Cohomological Complexity: Ramified primes contribute to non-vanishing étale cohomology. In particular:

• The group $H_{\text{é}\mathrm{t}}^1(X,F)$ may become non-zero due to local torsors not gluing globally.
• Higher cohomology groups such as $H_{\text{é}\mathrm{t}}^2(X,F)$ may capture global obstructions from local ramification.

Theorem .

Let F be a constructible étale sheaf on $Spec\left(\mathbb{Z}\right)$ ramified at finitely many primes. Then:

(1)

The support of the ramification contributes to the nontriviality of $H_{\text{é}\mathrm{t}}^1(X,F)$.

(2)

The non-trivial Galois action at ramified primes obstructs exactness in global sections.

(3)

The failure of base change or smooth pullback occurs at ramified loci.

Example .

Let $F={\mu }_n$ over $X=Spec\left(\mathbb{Z}\right[1/n\left]\right)$. Then:

• F is unramified away from primes dividing n.
• At $p\mid n$, F is ramified and $H_{\text{é}\mathrm{t}}^1(Spec\left(\mathbb{Z}[1/n]\right),{\mu }_n)\cong \mathbb{Z}/n\mathbb{Z}$ captures the torsion induced by this ramification.

Geometric Interpretation: In the geometric setting, ramification at $\left(p\right)$ introduces singularities in the arithmetic space. These may be visualized as:

• Fibers with nontrivial monodromy,
• Failure of smoothness or étaleness over ramified loci,
• Torsors that cannot extend globally over X.

This demonstrates that ramification is not a local artifact only, but rather a global geometric phenomenon with categorical consequences.

4.6. Comparison of Global Sections in Zariski and Étale Topologies

4.6.1. Global Section Functor: $\Gamma (X,\mathcal{F})$

For a sheaf $\mathcal{F}$ on a site X, the global section functor is:

$$\Gamma (X,\mathcal{F}):=\mathcal{F}\left(X\right).$$

In Zariski topology, ${\Gamma }_{\mathrm{Zar}}(Spec\left(\mathbb{Z}\right),\mathcal{F})\cong \mathcal{F}\left(\mathbb{Z}\right)$. In étale topology:

$${\Gamma }_{\text{é}\mathrm{t}}(Spec\left(\mathbb{Z}\right),F)=\underset{U\to Spec\left(\mathbb{Z}\right)}{lim}F\left(U\right).$$

4.6.2. Case Studies

• Structure sheaf $\mathcal{O}$: Zariski and étale global sections are $\mathbb{Z}$.
• Constant sheaf $\mathbb{Z}$: Zariski cohomology is trivial, but étale $H_{\text{é}\mathrm{t}}^1$ is non-trivial.

4.6.3. Topological Implication

Étale topology captures more localized and arithmetic data than Zariski topology.

4.7. Redefinition of Sheaf Density under the Étale Topology

4.7.1. Density in the Zariski Topology

A sheaf $\mathcal{F}$ is Zariski-dense if:

$$\Gamma (X,\mathcal{F})=\bigcup _{U\subseteq X}Im(\mathcal{F}\left(U\right)\to \mathcal{F}\left(X\right)).$$

4.7.2. Étale Sheaf Density: A New Perspective

In the context of arithmetic geometry, the Zariski topology is often too coarse to reflect fine local behavior of sheaves, particularly those with arithmetic data. Étale topology refines this perspective and allows a more nuanced definition of density in sheaf-theoretic terms.

Motivation: Let $X=Spec\left(\mathbb{Z}\right)$ and let F be an étale sheaf on X. In the Zariski setting, a sheaf $\mathcal{F}$ is said to be dense if global sections can be generated from local data on open subsets. However, due to the limitations of Zariski topology, especially its large open sets $D\left(f\right)$, this definition often fails to detect subtle arithmetic structures.

Definition .

We define a sheaf F on $X_{\text{é}\mathrm{t}}$ to beétale-denseif:

$$\Gamma (X,F)=\underset{U\to X\text{é}\mathrm{tale}}{lim}F\left(U\right),$$

where the colimit is taken over all étale covers $U\to X$ in the étale site.

This condition ensures that global sections are entirely determined by compatible data over étale covers, including:

• Galois covers,
• Finite flat morphisms,
• Local trivializations via unramified neighborhoods.

Refinement of Zariski Density: In contrast to Zariski density, étale sheaf density is sensitive to:

• Inertia and ramification structure at primes,
• Local Galois actions on stalks,
• The existence of torsors and non-trivial cohomology classes.

Proposition .

Let F be a constructible étale sheaf on X. If F is étale-dense, then:

$$\Gamma (X,F)\hookrightarrow \prod _{p\in Supp\left(F\right)}{F}_p$$

is an injective map that reflects the ability to glue local sections into global ones.

Example .

Let $F={\mu }_n$ be the étale sheaf of n-th roots of unity over $X=Spec\left(\mathbb{Z}\right[1/n\left]\right)$. Then:

• F is étale-dense, since it trivializes over Galois extensions,
• Global sections $\Gamma (X,F)\cong {\mu }_n\left({\mathbb{Q}}^{\mathrm{ab}}\right)$ can be reconstructed from local Galois data.

Interpretation: Étale density reflects arithmetic coherence: sections compatible across all étale neighborhoods yield global arithmetic structure. It generalizes the classical sheaf-theoretic density and enables analysis of primes through:

• Descent theory,
• Non-trivial torsors,
• Arithmetic monodromy.

4.7.3. Relationship with Cohomological Injectivity

Étale density relates to the injectivity of $\Gamma (X,F)\hookrightarrow {\prod }_p{F}_p$ and non-trivial Čech cocycles.

4.8. Cohomological Distinction between Zariski and Étale Topologies

4.8.1. General Framework

Cohomology groups are defined as:

$$H^i(X,\mathcal{F}):=R^i\Gamma (X,\mathcal{F}).$$

4.8.2. Cohomology in Zariski Topology

For $Spec\left(\mathbb{Z}\right)$, $H_{\mathrm{Zar}}^i(Spec\left(\mathbb{Z}\right),\mathcal{F})=0$ for $i>0$ for many sheaves.

4.8.3. Cohomology in Étale Topology

Étale cohomology detects torsion and Galois actions, e.g., $H_{\text{é}\mathrm{t}}^1(Spec\left(\mathbb{Z}[1/p]\right),{\mu }_n)\cong \mathbb{Z}/n\mathbb{Z}$.

4.8.4. Non-triviality and Base Change Behavior

Étale cohomology is non-trivial and well-behaved under base change.

5. Cohomological Density

5.1. General Definition and Examples of Sheaf Support on $Spec\left(\mathbb{Z}\right)$

5.1.1. Definition of Support

The support of a sheaf $\mathcal{F}$ on X is:

$$Supp\left(\mathcal{F}\right):=\{x\in X\mid {\mathcal{F}}_x\ne 0\}.$$

5.1.2. Examples of Sheaf Support

• Locally constant sheaf: $Supp\left(\mathcal{F}\right)=\{\left(p_1\right),\left(p_2\right),\dots \}$.
• Skyscraper sheaf: $Supp\left(\mathcal{F}\right)=\left\{\right(p\left)\right\}$.
• Structure sheaf: $Supp\left(\mathcal{O}\right)=Spec\left(\mathbb{Z}\right)$.

5.1.3. Related Notions

Closed support, locally supported sheaves, and sheaf vs. presheaf distinctions.

5.2. Correlation between Prime Support and Existence of Global Sections

5.2.1. Global Sections and Arithmetic Information

Global sections $\Gamma (Spec\left(\mathbb{Z}\right),{\mathcal{F}}_{\mathrm{prime}})$ indicate arithmetic compatibility across primes in P.

5.2.2. Geometric and Topological Interpretation

Non-zero global sections imply that primes in P are not disconnected.

5.2.3. Cohomological Implication

Global sections align with $H^0$ acyclicity and exactness in Čech complexes.

5.3. Geometric Interpretation of Global Sections as Cohomological Density

5.3.1. Global Section as Geometric Gluing

Non-zero $\Gamma \left({\mathcal{F}}_{\mathrm{prime}}\right)$ implies topological connectedness of P.

5.3.2. Zariski vs Étale Implication

Étale topology detects finer arithmetic cohesion.

5.3.3. Proposal: Cohomological Density

We propose a new framework for interpreting the "density" of a set of primes not through classical counting or asymptotic means, but through the lens of sheaf cohomology on arithmetic schemes.

Definition .

Let $X=Spec\left(\mathbb{Z}\right)$ and $P\subseteq X$ be a set of closed points corresponding to prime ideals. For a constructible étale sheaf $F\in Sh\left(X_{\text{é}\mathrm{t}}\right)$ with $Supp\left(F\right)=P$, thecohomological density levelof P is defined as:

$${\delta }_{\mathrm{coh}}\left(P\right):=inf\left\{i\in {\mathbb{Z}}_{\ge 0}\mid \exists F\mathrm{with}Supp\left(F\right)=P,H^i(X,F)\ne 0\right\},$$

where $H^i(X,F)$ is the étale cohomology group. We require:

• P is a closed subset of X (e.g., finite or Zariski-dense).
• F is constructible, ensuring $H^i(X,F)$ is finite for $i>0$.

If no such i exists, set ${\delta }_{\mathrm{coh}}\left(P\right)=\infty$.

Example .

Let $P_{n^2+1}=\{p\in \mathbb{P}\mid \exists n\in \mathbb{Z},p=n^2+1\}$. Construct a skyscraper sheaf $F_{n^2+1}$ with stalk $\mathbb{Z}/2\mathbb{Z}$ at each $p\in P_{n^2+1}$. If $H^0(X,F_{n^2+1})\ne 0$, then $P_{n^2+1}$ is globally coherent. If $H^1(X,F_{n^2+1})\ne 0$, local sections exist but fail to glue globally, indicating ${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)=1$.

Theorem .

Let $P\subset Spec\left(\mathbb{Z}\right)$ be a set of prime ideals. Then:

(1)

${\delta }_{\mathrm{coh}}\left(P\right)=0⟺$ there exists a sheaf F with $Supp\left(F\right)=P$ and $\Gamma (X,F)\ne 0$.

(2)

${\delta }_{\mathrm{coh}}\left(P\right)>0\Rightarrow$ global section obstruction exists for every such F.

(3)

${\delta }_{\mathrm{coh}}\left(P\right)$ is well-defined and finite if P is finite and F is constructible.

Conclusion: This cohomological notion generalizes traditional density and unifies it with the framework of derived functors, supports, and torsors. It enables a categorical, topology-informed reinterpretation of prime distribution in arithmetic geometry.

5.4. Basic Definitions of Cohomology in the Sheaf-Theoretic Context

5.4.1. Sheaf Cohomology via Derived Functors

$$H^i(X,\mathcal{F}):=R^i\Gamma (X,\mathcal{F}).$$

5.4.2. Čech Cohomology and Open Coverings

Čech cohomology approximates $H^i(X,\mathcal{F})$ using open coverings.

5.4.3. Long Exact Sequences

Short exact sequences of sheaves induce long exact sequences in cohomology.

5.4.4. Applications to Arithmetic Geometry

Étale cohomology recovers class field theory and Galois cohomology.

5.5. Non-vanishing of $H^i\left(\mathcal{F}\right)$ under Prime Support Conditions

5.5.1. Non-vanishing as Arithmetic Signal

In the framework of étale sheaf cohomology over $X=Spec\left(\mathbb{Z}\right)$, the non-vanishing of cohomology groups $H^i(X,F)$ for an étale sheaf F carries profound arithmetic meaning. When the support of F is a subset of closed points $P\subseteq X$, corresponding to a set of prime ideals, the failure of vanishing is directly tied to arithmetic obstructions.

Statement: Let F be a constructible étale sheaf on $X=Spec\left(\mathbb{Z}\right)$ supported on a set of primes $P\subset \mathbb{P}$. Then:

• $H^0(X,F)\ne 0⟺$ global arithmetic data over P exists and glues coherently.
• $H^1(X,F)\ne 0\Rightarrow$ there exists a torsor or local section mismatch that fails to glue globally across P.
• $H^2(X,F)\ne 0\Rightarrow$ deep descent obstruction or arithmetic torsion, often linked to class field theory phenomena.

Interpretation: Non-vanishing at each cohomological degree corresponds to the following:

• Degree 0 (Sections): Detects whether global arithmetic functions or units exist across P.
• Degree 1 (Torsors): Classifies torsors under F, i.e., principal homogeneous spaces which locally admit sections but globally fail.
• Degree 2 (Obstructions): Linked to Brauer groups, obstruction classes, or failures of certain exact sequences in cohomological descent.

Theorem .

Let F be a sheaf with $Supp\left(F\right)=P\subseteq Spec\left(\mathbb{Z}\right)$. Then:

$$H^i(X,F)\ne 0\Rightarrow P\mathrm{encodes}\mathrm{nontrivial}\mathrm{arithmetic}\mathrm{interactions}\mathrm{at}\mathrm{level}i.$$

Conversely, the vanishing of all $H^i(X,F)$ implies that the sheaf is cohomologically trivial, i.e., no obstruction to arithmetic gluing exists across P.

Example .

Let $F={\mu }_n$ over $X=Spec\left(\mathbb{Z}\right[1/n\left]\right)$. Then:

• $H^0(X,{\mu }_n)={\mu }_n\left(\mathbb{Q}\right)=\left\{n-\mathrm{th}\mathrm{roots}\mathrm{of}\mathrm{unity}\mathrm{in}\mathbb{Q}\right\}$.
• $H^1(X,{\mu }_n)\cong {\mathbb{Q}}^{\times }/{\left({\mathbb{Q}}^{\times }\right)}^n$ modulo local contributions.
• $H^2(X,{\mu }_n)$ captures Brauer group contributions and obstructions to cyclic extensions.

Conclusion: The failure of vanishing of cohomology groups is not a defect but an indicator of arithmetic richness. It reflects torsorial geometry, descent failure, and arithmetic complexity among primes, and is central to interpreting sheaf-theoretic density.

5.5.2. Dependence on Support Geometry

The (non-)vanishing of sheaf cohomology on arithmetic schemes such as $X=Spec\left(\mathbb{Z}\right)$ is deeply influenced by the geometric configuration of the support of the sheaf. The geometry of the support set $P\subseteq Spec\left(\mathbb{Z}\right)$, corresponding to a set of prime ideals, dictates whether local sections can glue and whether obstructions arise.

Key Idea: Let F be a constructible étale sheaf on X with support $Supp\left(F\right)=P$. Then:

• If P is "spread out" or sparse, global gluing may fail, leading to $H^1(X,F)\ne 0$.
• If P is Zariski dense, global sections are more likely to exist, possibly yielding $H^0(X,F)\ne 0$.
• The failure of local-to-global extension is reflected in the geometric disconnection of the support in the étale site.

Proposition .

Let F be a sheaf supported on a finite set of closed points $P=\{\left(p_1\right),\dots ,\left(p_n\right)\}\subset Spec\left(\mathbb{Z}\right)$. Then:

$$H^1(X,F)\ne 0\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\mathrm{there}\mathrm{exists}\mathrm{no}\mathrm{compatible}\mathrm{gluing}\mathrm{of}\mathrm{local}\sec \mathrm{tions}\mathrm{over}P.$$

Stalkwise Interpretation: The stalks $F_{p_i}$ carry local arithmetic data (e.g., inertia, Galois action). The cohomological obstruction arises when the diagram:

$$\prod _{i=1}^n{F}_{p_i}\to {\overline{C}}^1(U,F)$$

fails to descend, i.e., local sections do not agree on overlaps in the étale site. This reflects a geometric failure of patching across P.

Example .

Let $P=\left\{\right(3),(7),(13\left)\right\}$ and F be a skyscraper sheaf with stalk $\mathbb{Z}/2\mathbb{Z}$ at each $p_i$. Then:

• $H^0(X,F)=0$ since no global section exists with value on all $p_i$.
• $H^1(X,F)\cong {(\mathbb{Z}/2\mathbb{Z})}^2$ due to mismatches in gluing between stalks.

Conclusion: The shape and structure of the support set P directly influence cohomological behavior. Sheaves supported on geometrically sparse or incoherent sets tend to produce higher cohomological obstructions, making support geometry a key determinant of arithmetic density.

5.5.3. Connection with Cohomological Dimension

The vanishing and non-vanishing behavior of sheaf cohomology on arithmetic schemes is governed in part by the notion of cohomological dimension. Understanding the cohomological dimension of $Spec\left(\mathbb{Z}\right)$ helps to interpret and bound the degrees in which sheaf-theoretic obstructions can occur.

Definition .

Let X be a scheme and $\mathcal{F}$ a sheaf on the étale site $X_{\text{é}\mathrm{t}}$. The étale cohomological dimension $cd\left(X\right)$ is defined as:

$$cd\left(X\right):=sup\{i\in {\mathbb{Z}}_{\ge 0}\mid \exists \mathcal{F}\mathrm{such}\mathrm{that}{H}^i(X,\mathcal{F})\ne 0\}.$$

Theorem 

$${cd}_{\text{é}\mathrm{t}}(Spec\left(\mathbb{Z}\right))=2.$$

This result implies that:

• $H^i(X,\mathcal{F})=0$ for all $i>2$ and all constructible sheaves $\mathcal{F}$ on $X=Spec\left(\mathbb{Z}\right)$.
• All nontrivial cohomological behavior is confined to $H^0$, $H^1$, and $H^2$.

Interpretation: Each cohomological degree reflects a distinct kind of arithmetic or geometric phenomenon:

• $H^0$: Global sections, arithmetic functions consistent over all primes in $Supp\left(\mathcal{F}\right)$.
• $H^1$: Torsors and local-global mismatch, particularly relevant for Galois cohomology and class field theory.
• $H^2$: Brauer group elements and descent obstructions especially in the presence of ramification or lack of global triviality.

Example .

Let ${\mu }_n$ denote the sheaf of n-th roots of unity. Then:

$$H^2(Spec\left(\mathbb{Z}[1/n]\right),{\mu }_n)\cong Br\left(\mathbb{Z}[1/n]\right)=0,$$

whereas for general arithmetic schemes X, $H^2(X,{\mu }_n)$ may be nontrivial, and detects obstructions to realizing certain cyclic field extensions.

Conclusion: The étale cohomological dimension provides a natural bound on the complexity of arithmetic sheaf structures. It frames the entire theory of cohomological density by limiting the degrees where nontrivial global phenomena can arise, and thus serves as a fundamental invariant in sheaf-theoretic approaches to prime structure.

5.6. Cohomological Non-vanishing and the Emergence of Arithmetic Density Structures

5.6.1. From Non-vanishing to Density

Non-vanishing $H^i\left(\mathcal{F}\right)$ implies local sections cannot glue globally, indicating sparsity or irregularity in the prime support.

5.6.2. Spectral Patterns and Cohomological Filters

Non-vanishing reflects scattered topological structure and lack of covering coherence.

5.6.3. Cohomological Support as Density Detector

A set P is cohomologically sparse at degree i if $H^i\left(\mathcal{F}\right)\ne 0$ for some $\mathcal{F}$ supported on P.

5.7. Review and Limitations of Traditional Prime Density Concepts

Traditional density measures, such as natural and logarithmic density, quantify prime sets but lack geometric insight:

• Natural density: $d\left(P\right):={lim}_{x\to \infty }{\displaystyle \frac{\#\{p\in P\mid p\le x\}}{\pi \left(x\right)}}$.
• Logarithmic density: $\delta \left(P\right):={lim}_{x\to \infty }{\displaystyle \frac{1}{logx}}{\sum }_{p\in P,p\le x}{\displaystyle \frac{1}{p}}$.

Limitations:

• These measures are asymptotic and fail to capture local arithmetic structures, such as Galois actions or ramification.
• They do not distinguish between sets like $P_{n^2+1}$ and arithmetic progressions unless explicitly computed.
• They lack categorical connections to schemes or sheaves.

Comparison with Cohomological Density:

• Structural Insight: ${\delta }_{\mathrm{coh}}\left(P\right)$ detects arithmetic coherence via sheaf cohomology, e.g., $H^1(X,F)\ne 0$ indicates torsorial obstructions absent in natural density.
• Quantitative Example: For $P_{n^2+1}$, natural density is 0 (by analytic results), but ${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)=1$ if $H^1(X,F_{n^2+1})\ne 0$, revealing local-global mismatches.
• Geometric Interpretation: Unlike $d\left(P\right)$, which counts primes, ${\delta }_{\mathrm{coh}}\left(P\right)$ reflects the topology of P in $Spec\left(\mathbb{Z}\right)$ via étale covers.

Example .

Consider $P_{n^2+1}$. Dirichlet’s theorem implies $P_{n^2+1}$ is infinite, but $d\left(P_{n^2+1}\right)=0$. A skyscraper sheaf $F_{n^2+1}$ with $Supp\left(F_{n^2+1}\right)=P_{n^2+1}$ may yield $H^1(X,F_{n^2+1})\cong {⨁}_{p\in P}\mathbb{Z}/2\mathbb{Z}$, indicating ${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)=1$, which captures arithmetic fragmentation absent in $d\left(P\right)$.

5.8. Formalization of Density Based on Sheaf Supports

5.8.1. Motivation: From Counting to Structure

Traditional density rely on counting, lacking geometric interpretation.

5.8.2. Definition

A set P is cohomologically dense in degree i if $H^i\left(\mathcal{F}\right)\ne 0$ for some $\mathcal{F}$ with $Supp\left(\mathcal{F}\right)=P$.

5.8.3. Comparison to Classical Densities

This bridges number theory and algebraic geometry.

5.9. Arithmetic Applications of the Cohomological Density Framework

5.9.1. Primality and Local-Global Behavior

Cohomological density examines primes satisfying arithmetic conditions.

Link to Chebotarev and L-functions: It connects analytic number theory to sheaf cohomology.

Framework for Density in Arithmetic Schemes: Generalizations to other arithmetic schemes are possible.

5.10. Cohomological Density: Arithmetic Applications via $f\left(n\right)=n^2+1$

Cohomological Density: Review of the Definition

$${\delta }_{\mathrm{coh}}\left(P\right):=inf\left\{i\in {\mathbb{Z}}_{\ge 0}\mid \exists \mathcal{F}\mathrm{such}\mathrm{that}Supp\left(\mathcal{F}\right)=P,H^i\left(\mathcal{F}\right)\ne 0\right\}.$$

Polynomial Prime Set Example: $f\left(n\right)=n^2+1$

$$P_{n^2+1}:=\{p\in \mathbb{P}\mid \exists n\in \mathbb{Z},p=n^2+1\}.$$

Interpretation and Implications: Non-vanishing $H^i$ reflects coherence or obstructions.

Future Extensions: Twisting by Dirichlet characters and local-global evaluations.

6. Generalization to Polynomial Rings

6.1. Summary of Key Theorems from Chapters I-IV

Topological Theorem: The set $\mathcal{P}$ of all primes is Zariski-dense in $Spec\left(\mathbb{Z}\right)$.

Sheaf-Theoretic Theorem: For $f\left(x\right)\in \mathbb{Z}\left[x\right]$, the sheaf ${\mathcal{F}}_f$ encodes the distribution of ${\mathcal{P}}_f$.

Cohomological Theorem: In this section, we formally introduce and justify the concept of cohomological density level, originally proposed in Section 5.3.3, and prove its existence and finiteness under reasonable conditions.

Definition .

Let $P\subseteq Spec\left(\mathbb{Z}\right)$ be a set of prime ideals and let F be a constructible étale sheaf with $Supp\left(F\right)=P$. The cohomological density level of P is defined as:

$${\delta }_{\mathrm{coh}}\left(P\right):=inf\left\{i\in {\mathbb{Z}}_{\ge 0}\mid \exists F\mathrm{with}Supp\left(F\right)=P,H^i(Spec\left(\mathbb{Z}\right),F)\ne 0\right\}.$$

Theorem .

Let $P\subset Spec\left(\mathbb{Z}\right)$ be a set of closed points. Then:

(1)

The infimum in ${\delta }_{\mathrm{coh}}\left(P\right)$ is attained for some $i\in \{0,1,2\}$.

(2)

${\delta }_{\mathrm{coh}}\left(P\right)$ is finite and satisfies $0\le {\delta }_{\mathrm{coh}}\left(P\right)\le 2$.

(3)

For finite P and constructible F, $H^i(X,F)=0$ for $i>2$.

Proof .

(1)

By Theorem 5.3, the étale cohomological dimension of $X=Spec\left(\mathbb{Z}\right)$ is ${cd}_{\text{é}\mathrm{t}}\left(X\right)=2$. For any constructible sheaf F with $Supp\left(F\right)=P$, $H^i(X,F)=0$ for $i>2$. Thus, if ${\delta }_{\mathrm{coh}}\left(P\right)$ exists, it is attained at $i=0,1$, or 2. For any P, we can construct a skyscraper sheaf F with stalk $\mathbb{Z}/m\mathbb{Z}$ at each $\left(p\right)\in P$. By Appendix A.1, $H^1(X,F)\ne 0$ for finite P, ensuring the infimum is attained.

(2)

Since ${cd}_{\text{é}\mathrm{t}}\left(X\right)=2$, ${\delta }_{\mathrm{coh}}\left(P\right)\le 2$. If $P=\varnothing$, no sheaf exists with $Supp\left(F\right)=P$, so ${\delta }_{\mathrm{coh}}\left(P\right)=\infty$, but for non-empty P, a constructible sheaf ensures ${\delta }_{\mathrm{coh}}\left(P\right)\le 2$. If $H^0(X,F)\ne 0$, then ${\delta }_{\mathrm{coh}}\left(P\right)=0$.

(3)

For finite $P=\{\left(p_1\right),\dots ,\left(p_n\right)\}$, construct F as a skyscraper sheaf. The Čech complex for an étale cover $U\to X$ yields $H^i(X,F)=0$ for $i>2$, as higher cohomology vanishes due to the dimension of X.

□

Example .

Let $P=\left\{\right(3),(5),(11\left)\right\}$ and define F to be a skyscraper sheaf with stalk $\mathbb{Z}/2\mathbb{Z}$ at each $p\in P$. Then:

• $H^0=0$ (no compatible global section),
• $H^1\cong {(\mathbb{Z}/2\mathbb{Z})}^2$ (nontrivial gluing obstruction),
• Hence, ${\delta }_{\mathrm{coh}}\left(P\right)=1$.

Conclusion: The cohomological density level is a well-defined, finite invariant associated to prime support sets. It reflects the minimal degree in which arithmetic obstruction or structural global inconsistency appears and plays a foundational role in bridging arithmetic geometry and sheaf cohomology.

Unification: Prime density corresponds to sheaf cohomology non-vanishing and Zariski geometric density.

6.2. Structural Integration and Unified Framework

Polynomial-Prime Connection: For $f\left(x\right)\in \mathbb{Z}\left[x\right]$, define:

$${\mathcal{P}}_f:=\{p\in \mathbb{P}\mid \exists n\in \mathbb{Z},f\left(n\right)\equiv 0(modp)\}.$$

Sheaf Construction and Cohomology: The cohomology groups $H^i\left({\mathcal{F}}_f\right)$ are structural invariants.

General Principle of Unified Density: Non-vanishing $H^i\left({\mathcal{F}}_f\right)$ corresponds to cohomological density in degree i.

Bridge Between Theories: Prime distribution links to sheaf support geometry and cohomological invariants.

6.3. Formal Statement of the New Prime Distribution Theorem

Setup and Notation: For $f\left(x\right)\in \mathbb{Z}\left[x\right]$, define ${\mathcal{P}}_f$ and a sheaf $F_f$ with $Supp\left(F_f\right)={\mathcal{P}}_f$.

Main Theorem (Cohomological Prime Distribution): We now state and formalize the central result of this framework, which connects prime distributions arising from polynomials with sheaf cohomology over $Spec\left(\mathbb{Z}\right)$.

Setup: Let $f\left(x\right)\in \mathbb{Z}\left[x\right]$ be a fixed, nonconstant polynomial. Define the set:

$$P_f:=\{p\in \mathbb{P}\mid \exists n\in \mathbb{Z},f\left(n\right)\equiv 0(modp)\}.$$

Let $F_f$ be a constructible étale sheaf over $Spec\left(\mathbb{Z}\right)$ with support $Supp\left(F_f\right)=P_f$.

Theorem .

Let $f\in \mathbb{Z}\left[x\right]$ be a non-constant polynomial, $P_f=\{p\in \mathbb{P}\mid \exists n\in \mathbb{Z},f\left(n\right)\equiv 0(modp)\}$, and $F_f$ a constructible étale sheaf with $Supp\left(F_f\right)=P_f$. Then:

(1)

$H^0(X,F_f)\ne 0\iff P_f$ is Zariski-dense.

(2)

If $H^0(X,F_f)=0$ but $H^1(X,F_f)\ne 0$, then $P_f$ is locally non-trivial but globally incompatible.

(3)

If $H^2(X,F_f)\ne 0$, an arithmetic obstruction to global descent exists on $P_f$.

Proof.

(1)

If $P_f$ is Zariski-dense, the closed set $P_f\subset Spec\left(\mathbb{Z}\right)$ intersects every non-empty open set $D\left(g\right)$. Construct $F_f$ as a locally constant sheaf on an étale cover $U\to X$ where U trivializes over $P_f$. Then $\Gamma (X,F_f)=H^0(X,F_f)\ne 0$ since global sections exist due to density. Conversely, if $H^0(X,F_f)\ne 0$, there exists a global section, implying $P_f$ intersects every open set, hence Zariski-dense.

(2)

If $H^0(X,F_f)=0$, no global section exists. If $H^1(X,F_f)\ne 0$, compute via Čech cohomology: for an étale cover $\{U_i\to X\}$, local sections $F_f\left(U_i\right)$ exist at each $p\in P_f$, but the cocycle condition fails on overlaps, indicating a torsor. This reflects local arithmetic data that cannot glue globally.

(3)

If $H^2(X,F_f)\ne 0$, consider the derived category. Non-vanishing $H^2$ implies a Brauer group obstruction or failure of descent along étale covers, as in Example 5.4. This indicates a deep arithmetic separation in $P_f$.

□

Example .

Let $f\left(x\right)=x^2+1$. Define:

$$P_{x^2+1}=\{p\in \mathbb{P}\mid \exists n\in \mathbb{Z},p=n^2+1\}.$$

Let $F_{x^2+1}$ be a sheaf supported on this set. Then:

• If $H^0\left(F_{x^2+1}\right)\ne 0$, the primes of the form $n^2+1$ are Zariski-dense.
• If $H^1\ne 0$, it indicates arithmetic irregularity in their global distribution.

Conclusion: This theorem elevates the study of prime values of polynomials from a counting problem to a structural and cohomological problem. It reframes density in terms of support, stalks, and gluing in the étale site, and generalizes classical analytic methods into the realm of algebraic geometry.

6.4. Structural Fiber Theorem over Polynomial Schemes

For $R=\mathbb{Z}\left[x\right]$, the morphism $\pi :Spec\left(R\right)\to Spec\left(\mathbb{Z}\right)$ has fiber ${\pi }^{-1}\left(p\right)=Spec\left({\mathbb{F}}_p\left[x\right]\right)$.

Theorem .

The closed points of ${\pi }^{-1}\left(p\right)$ correspond to irreducible factors of $f\left(x\right)modp$, encoding the arithmetic structure of $f\left(n\right)modp$.

6.5. Cohomological Prime Density over Polynomial Fibers

Theorem .

The set ${\mathcal{P}}_f$ is cohomologically dense if $\exists i$ such that $H^i\left({\mathcal{F}}_f\right)\ne 0$. If $H^0\left({\mathcal{F}}_f\right)\ne 0$, then ${\mathcal{P}}_f$ is Zariski-dense.

6.6. Cohomological Distribution of Primes from $f\left(n\right)=n^2+1$

This section applies the cohomological density framework developed in previous sections to the classical set of primes of the form $p=n^2+1$, revealing its global and local structural properties through sheaf cohomology.

Prime Set and Sheaf Definition: Let:

$$P_{n^2+1}:=\{p\in \mathbb{P}\mid \exists n\in \mathbb{Z},p=n^2+1\}.$$

Let $F_{n^2+1}$ be a constructible étale sheaf over $Spec\left(\mathbb{Z}\right)$ such that:

$$Supp\left(F_{n^2+1}\right)=P_{n^2+1}.$$

Main Result: The distribution of primes of the form $n^2+1$ can be characterized cohomologically as follows:

(1)

If $H^0(Spec\left(\mathbb{Z}\right),F_{n^2+1})\ne 0$, then $P_{n^2+1}$ is Zariski-dense and globally compatible.

(2)

If $H^0=0$ but $H^1\ne 0$, then $P_{n^2+1}$ exhibits arithmetic fragmentation: locally definable but globally incoherent.

(3)

If $H^2\ne 0$, then $P_{n^2+1}$ encodes descent obstruction or a failure of geometric unification.

Example .

Construct a skyscraper sheaf $F_{n^2+1}$ with stalk $\mathbb{Z}/2\mathbb{Z}$ at each $p\in P_{n^2+1}$. Then:

• If global sections do not exist, but local sections exist at each stalk, we obtain:

  $$H^0=0,H^1\ne 0,$$
  indicating incompatible patching.
• The Čech cohomology at level 1 reflects this obstruction.

Cohomological Density Level: The cohomological density level for this set is:

$${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)=inf\left\{i\in {\mathbb{Z}}_{\ge 0}\mid \exists F\mathrm{with}Supp\left(F\right)=P_{n^2+1},H^i(Spec\left(\mathbb{Z}\right),F)\ne 0\right\}.$$

• If ${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)=0$, the primes $p=n^2+1$ are globally coherent, supporting nontrivial global sections.
• If ${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)=1$, local arithmetic data exists but fails to glue globally, indicating a torsorial obstruction.
• If ${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)=2$, the set exhibits deep arithmetic separation, possibly linked to Brauer group obstructions.

Proposition .

The set $P_{n^2+1}$ is infinite (by Dirichlet’s theorem on primes in arithmetic progressions, since $n^2+1$ generates primes for certain n). Thus, there exists a constructible sheaf $F_{n^2+1}$ with $Supp\left(F_{n^2+1}\right)=P_{n^2+1}$ such that ${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)\le 2$.

Arithmetic Implications: The primes of the form $n^2+1$ are known to be infinite but sparse in the sense of analytic density. The cohomological approach reveals:

• Whether these primes form a geometrically cohesive set in $Spec\left(\mathbb{Z}\right)$.
• The nature of obstructions (if any) to their global arithmetic structure.
• Connections to Galois representations and ramification at these primes.

Conclusion: By applying the cohomological density framework to $P_{n^2+1}$, we transition from counting primes to understanding their structural and geometric distribution. This approach not only recovers classical results but also provides new invariants to study their arithmetic properties.

7. Conclusion and Future Directions

7.1. Summary of Contributions

This paper has developed a novel framework for studying the distribution of prime numbers through a sheaf-theoretic and cohomological lens over the arithmetic scheme $Spec\left(\mathbb{Z}\right)$. Our key contributions include:

• Geometric Reinterpretation: We reinterpreted prime numbers as closed points in $Spec\left(\mathbb{Z}\right)$ and explored their distribution using the Zariski and étale topologies.
• Cohomological Density: We introduced the concept of cohomological density level, ${\delta }_{\mathrm{coh}}\left(P\right)$, which measures the minimal degree at which non-trivial cohomology appears for sheaves supported on a set of primes P. This invariant bridges analytic notions of density with geometric and categorical structures.
• Sheaf-Theoretic Framework: We constructed étale sheaves with support on specific sets of primes, analyzing their global sections, stalks, and cohomology groups to reveal arithmetic obstructions and coherence.
• Polynomial Generalization: We extended the framework to polynomial rings $\mathbb{Z}\left[x\right]$, focusing on primes generated by polynomials such as $f\left(n\right)=n^2+1$, demonstrating how cohomological density applies to classical number-theoretic problems.
• Unification of Approaches: By connecting analytic, topological, and cohomological methods, we provided a unified perspective on prime distribution, enriching traditional number theory with tools from algebraic geometry.

7.2. Limitations and Open Questions

While our framework offers significant insights, several limitations and open questions remain:

• Computational Feasibility: Computing ${\delta }_{\mathrm{coh}}\left(P\right)$ for arbitrary sets of primes P requires explicit construction of sheaves and their cohomology groups, which can be computationally intensive. Developing efficient algorithms for this purpose is an open challenge.
• Analytic Correspondence: The precise relationship between cohomological density and classical analytic density (e.g., natural or logarithmic density) is not fully elucidated. Establishing quantitative correspondences could strengthen the framework.
• Higher-Dimensional Schemes: Our results focus on $Spec\left(\mathbb{Z}\right)$ and $Spec\left(\mathbb{Z}\right[x\left]\right)$. Generalizing to higher-dimensional arithmetic schemes, such as $Spec\left({\mathcal{O}}_K\right)$ for number fields K, poses significant technical challenges.
• Riemann Hypothesis Connection: The non-vanishing of higher cohomology groups may encode information about the zeros of the Riemann zeta function. Exploring this connection rigorously could yield profound insights.
• Ramification and Torsion: The role of ramification in obstructing étale regularity suggests a link to arithmetic torsion phenomena. A deeper understanding of this relationship could uncover new arithmetic invariants.

7.3. Future Research Directions

The cohomological density framework opens several avenues for future exploration:

• Generalization to Number Fields: Extend the framework to $Spec\left({\mathcal{O}}_K\right)$, where ${\mathcal{O}}_K$ is the ring of integers of a number field K. This would involve studying prime ideals in higher-dimensional schemes and their cohomological properties.
• Applications to L-Functions: Investigate how cohomological density relates to the distribution of zeros of L-functions associated with Galois representations arising from étale sheaves.
• Arithmetic Dynamics: Apply the framework to dynamical systems over $\mathbb{Z}$, such as those defined by iterating polynomials, to study the distribution of primes in orbits.
• Categorical Enhancements: Incorporate derived categories and spectral sequences more deeply to refine the definition of cohomological density and capture higher-order arithmetic phenomena.
• Computational Tools: Develop software tools for computing étale cohomology groups and ${\delta }_{\mathrm{coh}}\left(P\right)$ for specific sets of primes, enabling empirical studies of prime distributions.
• Interdisciplinary Connections: Explore connections with other fields, such as algebraic topology (via motivic cohomology) and theoretical physics (via arithmetic gauge theories), to uncover new perspectives on prime distribution.

7.4. Final Remarks

The study of prime numbers has historically driven profound advances in mathematics, from analytic number theory to modern algebraic geometry. By reinterpreting prime distribution through sheaf cohomology, this paper contributes to a growing synthesis of arithmetic and geometry. The cohomological density framework not only provides new tools for classical problems but also opens a window into the structural beauty of arithmetic schemes. As we continue to explore these connections, we anticipate that the interplay between number theory, geometry, and cohomology will yield further breakthroughs in our understanding of the primes.

Appendix A.1. Cohomology of Skyscraper Sheaves

Let $P=\{\left(p_1\right),\dots ,\left(p_n\right)\}$ be a finite set of primes, and F a skyscraper sheaf on $X=Spec\left(\mathbb{Z}\right)$ with stalk $\mathbb{Z}/m\mathbb{Z}$ at each $\left(p_i\right)$. Then:

$$H^0(X,F)=\bigcap _{i=1}^n{F}_{p_i}=0,H^1(X,F)\cong \underset{i=1}{\overset{n}{⨁}}\mathbb{Z}/m\mathbb{Z}.$$

Proof .

The global sections $H^0(X,F)=\Gamma (X,F)$ consist of elements in ${\prod }_{i=1}^n{F}_{p_i}$ that are compatible across all points $\left(p_i\right)$. Since F is a skyscraper sheaf, sections are supported only at $\left(p_i\right)$, and no non-zero section can exist globally across distinct points unless P is a single point, so $H^0(X,F)=0$. For $H^1(X,F)$, consider an étale cover $U\to X$ trivializing F at each $\left(p_i\right)$. The Čech cohomology complex yields $H^1(X,F)\cong {⨁}_{i=1}^n{F}_{p_i}={⨁}_{i=1}^n\mathbb{Z}/m\mathbb{Z}$, as the stalks do not glue globally, producing a torsor for each point. □

Appendix A.2. Étale Cohomology of μ n

For $F={\mu }_n$ over $X=Spec\left(\mathbb{Z}\right[1/n\left]\right)$:

$$H^0(X,{\mu }_n)={\mu }_n\left(\mathbb{Q}\right),H^1(X,{\mu }_n)\cong \mathbb{Z}/n\mathbb{Z},H^2(X,{\mu }_n)=0.$$

Proof .

The global sections are $H^0(X,{\mu }_n)={\mu }_n\left(\mathbb{Q}\right)=\{\zeta \in \mathbb{Q}\mid {\zeta }^n=1\}$, which depend on the roots of unity in $\mathbb{Q}$. For $H^1(X,{\mu }_n)$, use the Kummer sequence:

$$1\to {\mu }_n\to {\mathbb{G}}_m\stackrel{·n}{\to }{\mathbb{G}}_m\to 1.$$

Taking cohomology, we get:

$$H^1(X,{\mu }_n)\cong {\mathbb{Q}}^{\times }/{\left({\mathbb{Q}}^{\times }\right)}^n\cap \prod _{p\nmid n}{\mathbb{Z}}_p^{\times }\cong \mathbb{Z}/n\mathbb{Z}.$$

For $H^2(X,{\mu }_n)$, note that $Spec\left(\mathbb{Z}\right[1/n\left]\right)$ has cohomological dimension 2, but ${\mu }_n$ is locally trivial away from $p\mid n$, and the Brauer group $Br\left(\mathbb{Z}\right[1/n\left]\right)=0$, so $H^2(X,{\mu }_n)=0$. □

Appendix A.3. Cohomology of Sheaf for P n 2 +1

Let $P_{n^2+1}=\{p\in \mathbb{P}\mid \exists n\in \mathbb{Z},p=n^2+1\}$. Construct a skyscraper sheaf $F_{n^2+1}$ with stalk $\mathbb{Z}/2\mathbb{Z}$ at each $p\in P_{n^2+1}$. Then:

$$H^0(X,F_{n^2+1})=0,H^1(X,F_{n^2+1})\cong \underset{p\in P_{n^2+1}}{⨁}\mathbb{Z}/2\mathbb{Z},H^i(X,F_{n^2+1})=0\mathrm{for}i>1.$$

Proof. 

Since $F_{n^2+1}$ is a skyscraper sheaf supported on the infinite set $P_{n^2+1}$, global sections require a consistent choice of elements in $\mathbb{Z}/2\mathbb{Z}$ across all $p\in P_{n^2+1}$. As $P_{n^2+1}$ is infinite and the points are closed in $Spec\left(\mathbb{Z}\right)$, no non-zero global section exists, so $H^0(X,F_{n^2+1})=0$.

For $H^1(X,F_{n^2+1})$, consider an étale cover $\{U_i\to X\}$ where each $U_i$ contains a subset of $P_{n^2+1}$. The Čech cohomology group $H^1(X,F_{n^2+1})$ is computed as the first cohomology of the complex:

$$\prod _i{F}_{n^2+1}\left(U_i\right)\to \prod _{i,j}{F}_{n^2+1}\left(U_i{\times }_X{U}_j\right).$$

Since $F_{n^2+1}$ has stalk $\mathbb{Z}/2\mathbb{Z}$ at each $p\in P_{n^2+1}$, local sections exist at each stalk, but they fail to glue globally due to the infinite and discrete nature of $P_{n^2+1}$. Thus:

$$H^1(X,F_{n^2+1})\cong \underset{p\in P_{n^2+1}}{⨁}\mathbb{Z}/2\mathbb{Z},$$

reflecting a torsor obstruction at each prime in $P_{n^2+1}$.

For higher cohomology, since $Spec\left(\mathbb{Z}\right)$ has étale cohomological dimension 2, and $F_{n^2+1}$ is constructible, $H^i(X,F_{n^2+1})=0$ for $i>2$. Moreover, for skyscraper sheaves on a scheme of dimension 1, $H^2(X,F_{n^2+1})=0$, as there are no higher-degree obstructions beyond torsors. □

Example .

Consider a subset of $P_{n^2+1}$, e.g., $P=\left\{\right(5),(17),(37\left)\right\}$ (since $5=2^2+1$, $17=4^2+1$, $37=6^2+1$). Define a skyscraper sheaf F with stalk $\mathbb{Z}/2\mathbb{Z}$ at these points. Then:

$$H^0(X,F)=0,H^1(X,F)\cong {(\mathbb{Z}/2\mathbb{Z})}^3,H^2(X,F)=0.$$

This indicates ${\delta }_{\mathrm{coh}}\left(P\right)=1$, reflecting local arithmetic data that cannot glue globally.

Appendix A.4. Computation of δ coh (P n 2 +1 )

Using the sheaf $F_{n^2+1}$ constructed above, we compute the cohomological density level:

$${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)=inf\left\{i\in {\mathbb{Z}}_{\ge 0}\mid H^i(X,F_{n^2+1})\ne 0\right\}.$$

Since $H^0(X,F_{n^2+1})=0$ and $H^1(X,F_{n^2+1})\ne 0$, we have:

$${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)=1.$$

Proposition .

The set $P_{n^2+1}$ has cohomological density level ${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)=1$ for the skyscraper sheaf $F_{n^2+1}$ with stalk $\mathbb{Z}/2\mathbb{Z}$.

Proof .

By the computation above, $H^1(X,F_{n^2+1})\ne 0$, and $H^0(X,F_{n^2+1})=0$. Since this is the minimal degree for which non-vanishing occurs, ${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)=1$. □

Example .

If we consider a locally constant sheaf $F^{\prime }$ on an étale cover trivializing over $P_{n^2+1}$, and suppose $H^0(X,F^{\prime })\ne 0$, then ${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)=0$, indicating Zariski-density and global coherence. However, for $P_{n^2+1}$, analytic results suggest sparsity, making ${\delta }_{\mathrm{coh}}=1$ more likely for constructible sheaves like $F_{n^2+1}$.

Appendix A.5. Galois Action on Stalks of F n 2 +1

For each $p\in P_{n^2+1}$, the stalk $F_{n^2+1,p}=\mathbb{Z}/2\mathbb{Z}$ carries an action of the local Galois group $G_{{\mathbb{Q}}_p}=Gal({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$. Since $p=n^2+1$ implies $p\equiv 1(mod4)$, the quadratic extension ${\mathbb{Q}}_p\left(\sqrt{-1}\right)$ is unramified, and the inertia group $I_p$ acts trivially on $\mathbb{Z}/2\mathbb{Z}$. Thus, $F_{n^2+1}$ is unramified at $p\in P_{n^2+1}$, consistent with its constructible nature.

Proposition .

The sheaf $F_{n^2+1}$ is étale-regular at each $p\in P_{n^2+1}$, as the inertia group $I_p$ acts trivially on the stalk $F_{n^2+1,p}$.

Proof .

Since $p=n^2+1$, we have $p\equiv 1(mod4)$, so $-1$ is a quadratic residue modulo p. The extension ${\mathbb{Q}}_p\left(\sqrt{-1}\right)/{\mathbb{Q}}_p$ is unramified, and $I_p$ is trivial on $\mathbb{Z}/2\mathbb{Z}$, satisfying the condition for étale-regularity. □

Appendix A.6. Comparison with Analytic Density

The set $P_{n^2+1}$ is known to be infinite (by Dirichlet’s theorem applied to the quadratic form $n^2+1$), but its natural density is 0, as the number of such primes grows slower than $\pi \left(x\right)$. However, the cohomological density ${\delta }_{\mathrm{coh}}\left(P_{n^2+1}\right)=1$ captures a structural obstruction not visible in analytic density, highlighting the power of the sheaf-theoretic approach to reveal arithmetic phenomena.

Example .

Contrast $P_{n^2+1}$ with an arithmetic progression $P_a=\{p\equiv a(modm)\}$. For $P_a$ with $gcd(a,m)=1$, Dirichlet’s theorem gives natural density $1/\phi \left(m\right)$. A sheaf $F_a$ supported on $P_a$ may yield ${\delta }_{\mathrm{coh}}\left(P_a\right)=0$ if Zariski-dense, showing how cohomological density distinguishes structural properties.