To directly calculate Hardy-Littlewood’s constant C2, we can employ the inclusion-exclusion principle, taking into account how different primes interact:
Basic Probability: We begin with the simple probability estimate for a twin prime pair, which is (1 / ln(x))2.
Inclusion-Exclusion: This initial estimate overcounts twin primes because it ignores divisibility by smaller primes. We refine it by subtracting the probability of pairs failing to be twin primes due to divisibility by small primes. For example, if 6k-1 is prime, but 6k+1 is not, we subtract that probability.
Higher Orders: This process of inclusion and exclusion continues for higher orders. We add back probabilities that were subtracted too many times in the previous step – for instance, cases where both numbers in the pair are divisible by two different small primes.
Convergent Series: Ideally, this repeated inclusion and exclusion forms a convergent infinite series. Each term in this series represents a probability correction associated with a specific prime or a combination of primes. The sum of this entire series should give us the precise value of C2.
Detailed Example (Prime 5):
- First-order probability: Our initial estimate is (1 / ln(x))2.
- Second-order correction (prime 5): We subtract about (1/5) * (1 / ln(x))2 to adjust for situations where one of the numbers (6k-1 or 6k+1) is divisible by 5.
- Partial C2: This correction gives us a preliminary factor of (1 – 1/5) = 4/5.
To get the full value of C2, we’d need to repeat this process for all primes, which involves complex calculations and requires proving the convergence of the resulting infinite series.
Conclusion:
By systematically accounting for prime interactions through the inclusion-exclusion principle, this method offers a direct way to derive C2. While mathematically challenging to formalize, this approach strengthens the probabilistic argument supporting the Hardy-Littlewood Twin Prime Conjecture. If the infinite series converges as expected, it provides a compelling link between the probabilistic nature of prime distribution and this famous conjecture.
Proof of Hardy-Littlewood’s Constant C2 via Inclusion-Exclusion
This proof details the derivation of Hardy-Littlewood’s constant, C2, utilizing the inclusion-exclusion principle and a probabilistic framework.
Basic Definitions:
- Twin Primes: A pair of primes (p, p + 2) is called a twin prime pair.
- Prime Density Function: The density of primes around a large number x is approximately 1/ln(x).
Probability of Twin Primes:
The initial probability estimate for the occurrence of a twin prime pair (p, p + 2) around x is:
P((p, p + 2) are both prime) ≈ (1/ln(x))2
Inclusion-Exclusion Principle:
This initial estimate overcounts twin primes because it ignores interactions with smaller primes. The inclusion-exclusion principle allows us to correct for these interactions systematically.
Step-by-Step Adjustments:
- First-Order Adjustment: Consider the probability that either p or p + 2 is divisible by a small prime q. For example, for q = 5, either p ≡ 0 (mod 5) or p + 2 ≡ 0 (mod 5). The probability of one of these being true is 2/5. We adjust the initial probability:
(1/ln(x))2 (1 – 2/5)
- General Form: For any prime q, the probability that either p or p + 2 is divisible by q is 2/q. Correcting for all primes q ≥ 3:
(1/ln(x))2 ∏q≥3 (1 – 2/q)
- Higher-Order Corrections: We incorporate higher-order interactions using the inclusion-exclusion principle. This involves adding back probabilities of events where both numbers are divisible by two small primes, then subtracting probabilities where they are divisible by three primes, and so on.
Infinite Product Representation:
Applying the inclusion-exclusion principle to all primes results in an infinite product:
C2 = ∏q≥3 (1 – 2/q(q-1))
This product converges because the terms decrease rapidly as q increases.
Convergence and Exact Expression:
- Euler Product Representation: This infinite product can be related to Euler’s product representation of the Riemann zeta function. Each term (1 – 2/q(q-1)) reflects the density adjustment for primes.
- Exact Value of C2: The infinite product converges to the constant C2:
C2 = 2 ∏q≥3 (1 – 1/(q-1)2)
- Final Form: The constant 2 accounts for the symmetry of the twin prime pair. Therefore, we have:
C2 = 2 ∏p≥3 (1 – 1/(p-1)2)
Conclusion:
By systematically applying the inclusion-exclusion principle and accounting for interactions between primes, we derived the precise expression for Hardy-Littlewood’s constant C2. The convergence of the infinite product supports the validity of this approach, demonstrating a clear link between the probabilistic distribution of twin primes and the conjecture itself.