Limit of Difference between Consecutive Prime Numbers

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Theorem

The Prime Number Theorem indicates that the average value of the difference between two consecutive prime numbers is of the order of $\log p_n$.

Let $E = \ds \liminf_{n \mathop \to \infty} \dfrac {p_{n + 1} - p_n} {\log p_n}$.

If there are infinitely many twin primes, then $E = 0$.

If not, then it is not known what the value of $E$ is.


Historical Note

Gregorio Ricci-Curbastro showed that $E \le \dfrac {15} {16}$.

Enrico Bombieri and Harold Davenport showed in $1966$ that $E \le \dfrac {2 + \sqrt 3} 8 \approx 0 \cdotp 46650 \ldots$

In their $1983$ work Les Nombres Remarquables, François Le Lionnais and Jean Brette report that Pil'Tai deduced in $1972$ that $E \le \dfrac {2 \sqrt 2 - 1} 4 \approx 0 \cdotp 45706 \ldots$

However, it is not clear who Pil'Tai is or was, and this has not been corroborated.

In $1973$ Martin Neil Huxley showed that $E \le \dfrac 1 4 + \dfrac \pi {16} \approx 0 \cdotp 44634 \, 95408 \ldots$


Sources