Limit of Difference between Consecutive Prime Numbers
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
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,44634 95408 \ldots$