The mysteries of twin primes: Relative density, heuristics, and analytical perspectives.

Author: Marcelo Fontinele (Collaborative Thesis Project)

1. Introduction

Twin primes — pairs of primes $(p, p+2)$ — play a central role in number theory, linking sieve techniques, heuristic analysis, and deep conjectures. Although the Twin Prime Conjecture (that infinitely many such pairs exist) remains unproven, modern developments (GPY methods, Zhang, Maynard–Tao, and sieve refinements) have reshaped our understanding of the distribution of prime gaps.

This work proposes a study focusing on:

1. Formalization and basic properties;

2. The notion of relative density in increasing intervals and among primes;

3. Hardy–Littlewood-type heuristics;

4. An overview of classical (Brun, Selberg, Chen) and modern results;

5. An analytical–computational plan for empirical verification.

2. Definitions and Notation

1. Twin Primes

   $$\pi_2(x) := \#\{\,p \le x : p,\; p+2 \;\text{are prime}\,\}.$$

2. Prime-Counting Function

   $$\pi(x) := \#\{p \le x\}.$$

3. Twin Prime Constant

   $$C_2 := \prod_{p\ge 3} \frac{p(p-2)}{(p-1)^2}.$$

4. Second-Order Logarithmic Integral

   $$\operatorname{Li}_2(x) := \int_{2}^{x} \frac{dt}{(\ln t)^2}.$$

Remark (6n ± 1 Form): Every prime $p>3$ satisfies $p \equiv \pm 1 \pmod 6$. Hence, except for $(3,5)$, every twin prime pair is of the form $(6n-1, 6n+1)$.

3. Densities: Natural, Relative, and Logarithmic

3.1. Natural Density Among Integers

$$\frac{\pi_2(x)}{x} \asymp \frac{1}{(\ln x)^2} \quad \xrightarrow[x\to\infty]{} 0.$$

Thus, the natural density of twin primes among integers is zero.

3.2. Relative Density Among Primes

Define:

$$\Delta\pi(x;h) := \pi(x+h) - \pi(x), \quad \Delta\pi_2(x;h) := \pi_2(x+h) - \pi_2(x).$$

The local relative density is

$$\mathcal{R}(x;h) := \frac{\Delta\pi_2(x;h)}{\Delta\pi(x;h)}.$$

Heuristically:

$$\mathcal{R}(x;h) \sim \frac{2C_2}{\ln x} \quad (x\to\infty).$$

This shows a slow decay of order $1/\ln x$.

3.3. Logarithmic Density

Brun’s constant:

$$B_2 := \sum_{\substack{p\;\text{twin}}} \left( \frac{1}{p} + \frac{1}{p+2} \right) < \infty.$$

Thus, even with logarithmic weighting, the density is zero.

4. Hardy–Littlewood Heuristic (Conjecture HL(2))

$$\boxed{\pi_2(x) \sim 2C_2 \, \operatorname{Li}_2(x)} \quad (x \to \infty).$$

Simplified form:

$$\pi_2(x) \sim \frac{2 C_2 \, x}{(\ln x)^2} \Big(1 + O\!\big(\tfrac{1}{\ln x}\big)\Big).$$

4.1. Local Factors

$$C_2 = \prod_{p\ge 3} \left(1 - \frac{1}{(p-1)^2}\right).$$

4.2. Predictions in Windows

For $h \ge (\ln x)^{2+\varepsilon}$:

$$\Delta\pi_2(x;h) \approx \frac{2 C_2\,h}{(\ln x)^2}.$$

A Poisson model with mean

$$\mu = \frac{2C_2 h}{(\ln x)^2}$$

predicts fluctuations.

5. Classic and Modern Results

• Brun (1919):
  $B_2 < \infty$.

• Selberg & Brun Sieves:
  $\pi_2(x) \ll x/(\ln x)^2.$

• Chen (1973):
  Infinitely many $p$ with $p+2$ prime or semiprime.

• GPY, Zhang, Maynard–Tao:
  Infinitely many bounded gaps between primes.

6. Relative Density in Intervals

6.1. Additive Windows

$$D_{\text{add}}(x;h) := \frac{\Delta\pi_2(x;h)}{h} \;\;\approx\;\; \frac{2C_2}{(\ln x)^2}.$$

6.2. Multiplicative Windows

For fixed $\theta \in (0,1]$:

$$\Delta\pi_2(x;\theta x) \approx 2C_2 \frac{\theta x}{(\ln x)^2}.$$ $$\mathcal{R}(x;\theta x) \approx \frac{2C_2}{\ln x}.$$

6.3. Congruence Restrictions

Local exclusions (e.g., modulo 3, 5, 7) yield the product $C_2$.

7. Heuristic Proposal (H*)

$$\pi_2(x) = 2C_2 \,\operatorname{Li}_2(x) + E(x),$$

with

$$E(x) = O\!\left(\frac{x}{(\ln x)^{2+\varepsilon}}\right).$$

Uniformly for $(\ln x)^{2+\varepsilon} \ll h \le x^{1-\varepsilon}$:

$$\Delta\pi_2(x;h) = \frac{2C_2 h}{(\ln x)^2} + O\!\left(\frac{h}{(\ln x)^{2+\varepsilon}}\right).$$

7.2. Evaluation Metrics

$$\rho(x) := \frac{\pi_2(x)}{2C_2 \,\operatorname{Li}_2(x)},$$ $$\varepsilon(x;h) := \frac{\Delta\pi_2(x;h)}{2C_2 h/(\ln x)^2} - 1.$$

8. Computational Plan

1. Segmented sieve up to $10^6, 10^7, 10^8$.

2. Compute $\pi_2(x)$, $\Delta\pi_2(x;h)$.

3. Fit regression to $\alpha x/(\ln x)^2$.

4. Test variance vs. Poisson predictions.

5. Visualization: curves of $\rho(x)$, heatmaps of $\varepsilon(x;h)$.

9. Limitations

• Sieve parity problem.

• Short intervals beyond current techniques.

• Dependence on probabilistic models.

10. Partial Conclusions

Twin primes have:

• Zero global density,

• Relative density decaying as

  $$\frac{1}{\ln x} \quad \text{(among primes)}, \qquad \frac{1}{(\ln x)^2} \quad \text{(among integers)}.$$

The HL(2) heuristic (Proposal H*) provides a consistent growth law, aligning classical sieves with modern prime-gap results.

11. Next Steps

1. Derivation of Hardy–Littlewood product.

2. Sieve methods: Brun, Selberg, parity problem.

3. Conditional results under GRH/Elliott–Halberstam.

4. Modern bounded gaps methods (GPY, Zhang, Maynard–Tao).

5. Computational appendix with algorithms and pseudocode.