Prime gap

Upper bounds

Bertrand's postulate, proven in 1852, states that there is always a prime number between ${\displaystyle k}$ and ${\displaystyle 2k}$, so in particular ${\displaystyle p_{n+1}<2p_{n}}$, which means ${\displaystyle g_{n}<p_{n}}$.

The prime number theorem, proven in 1896, says that the average length of the gap between a prime ${\displaystyle p}$ and the next prime will asymptotically approach ${\displaystyle \ln p}$, the natural logarithm of ${\displaystyle p}$, for sufficiently large primes. The actual length of the gap might be much more or less than this. However, one can deduce from the prime number theorem that the gaps get arbitrarily smaller in proportion to the primes: the quotient

${\displaystyle \lim _{n\to \infty }{\frac {g_{n}}{p_{n}}}=0.}$

In other words (by definition of a limit), for every ${\displaystyle \epsilon >0}$, there is a number ${\displaystyle N}$ such that for all ${\displaystyle n>N}$,

${\displaystyle g_{n}<p_{n}\epsilon }$.

Hoheisel (1930) was the first to show^[12] a sublinear dependence; that there exists a constant ${\displaystyle \theta <1}$ such that

${\displaystyle \pi (x+x^{\theta })-\pi (x)\sim {\frac {x^{\theta }}{\log x}}{\text{ as }}x\to \infty ,}$

hence showing that

${\displaystyle g_{n}\leqslant {p_{n}}^{\theta }}$

for sufficiently large ${\displaystyle n}$.

Hoheisel obtained the possible value 32999/33000 for ${\displaystyle \theta }$. This was improved to 249/250 by Heilbronn,^[13] and to ${\displaystyle \theta =3/4+\epsilon }$, for any ${\displaystyle \epsilon >0}$, by Chudakov.^[14]

A major improvement is due to Ingham,^[15] who showed that for some positive constant c,

if ${\displaystyle \zeta (1/2+it)=O(t^{c})}$ then ${\displaystyle \pi (x+x^{\theta })-\pi (x)\sim {\frac {x^{\theta }}{\log x}}}$ for any ${\displaystyle \theta >(1+4c)/(2+4c).}$

Here, O refers to the big O notation, ζ denotes the Riemann zeta function and π the prime-counting function. Knowing that any c > 1/6 is admissible, one obtains that θ may be any number greater than 5/8.

Since 5/8+ε < 2/3, and the gap between consecutive cubes is of the order of ${\displaystyle n^{2/3}}$, it follows that there is always a prime number between n³ and (n + 1)³, if n is sufficiently large.^[16] In 2016, Dudek gave an explicit version of Ingham's result: there are primes between consecutive cubes for all ${\displaystyle n>e^{e^{33.217}}\approx 1.01\cdot 10^{115809466034000}}$.^[17]

The Lindelöf hypothesis would imply that Ingham's formula holds for c any positive number: but even this would not be enough to imply that there is a prime number between n² and (n + 1)² for n sufficiently large (see Legendre's conjecture). To verify this, a stronger result such as Cramér's conjecture would be needed.

Huxley in 1972 showed that one may choose θ = 7/12 = 0.583.^[18]

A result, due to Baker, Harman and Pintz in 2001, shows that θ may be taken to be 0.525.^[19]

The above describes limits on all gaps; another area of interest is the minimum gap size. The twin prime conjecture asserts that there are always more gaps of size 2, but remains unproven. In 2005, Daniel Goldston, János Pintz and Cem Yıldırım proved that

${\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=0}$

and 2 years later improved this^[20] to

${\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{{\sqrt {\log p_{n}}}(\log \log p_{n})^{2}}}<\infty .}$

In 2013, Yitang Zhang proved that

${\displaystyle \liminf _{n\to \infty }g_{n}<7\cdot 10^{7},}$

meaning that there are infinitely many gaps that do not exceed 70 million.^[21] A Polymath Project collaborative effort to optimize Zhang's bound managed to lower the bound to 4680 on July 20, 2013.^[22] In November 2013, James Maynard introduced a new refinement of the GPY sieve, allowing him to reduce the bound to 600 and also show that the gaps between primes m apart are bounded for all m. That is, for any m there exists a bound Δ_m such that p_n+m − p_n ≤ Δ_m for infinitely many n.^[23] Using Maynard's ideas, the Polymath project improved the bound to 246;^[22][24] assuming the Elliott–Halberstam conjecture and its generalized form, the bound has been reduced to 12 and 6, respectively.^[22]

Lower bounds

In 1931, Erik Westzynthius proved that maximal prime gaps grow more than logarithmically. That is,^[1]

${\displaystyle \limsup _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=\infty .}$

In 1938, Robert Rankin proved the existence of a constant c > 0 such that the inequality

${\displaystyle g_{n}>{\frac {c\ \log n\ \log \log n\ \log \log \log \log n}{(\log \log \log n)^{2}}}}$

holds for infinitely many values of n, improving the results of Westzynthius and Paul Erdős. He later showed that one can take any constant c < e^γ, where γ is the Euler–Mascheroni constant. The value of the constant c was improved in 1997 to any value less than 2e^γ.^[25]

Paul Erdős offered a $10,000 prize for a proof or disproof that the constant c in the above inequality may be taken arbitrarily large.^[26] This was proved to be correct in 2014 by Ford–Green–Konyagin–Tao and, independently, James Maynard.^[27][28]

The result was further improved to

${\displaystyle g_{n}>{\frac {c\ \log n\ \log \log n\ \log \log \log \log n}{\log \log \log n}}}$

for infinitely many values of n by Ford–Green–Konyagin–Maynard–Tao.^[29]

In the spirit of Erdős' original prize, Terence Tao offered US$10,000 for a proof that c may be taken arbitrarily large in this inequality.^[30]

Lower bounds for chains of primes have also been determined.^[31]