15 Distribution of primes: short ranges
Recall that \(\Lambda \) is the von Mangoldt function, and that the prime number theorem asserts that
for unbounded \(x\). If \(p_n\) denotes the \(n^{\mathrm{th}}\) prime, the prime number theorem is also equivalent to
for unbounded \(n\).
We now consider local versions of the prime number theorem.
We let \(\theta _{\mathrm{PNT}}\) denote the least exponent with the following property: if \(\varepsilon {\gt} 0\) is fixed, and \(x\) is unbounded, then
\[ \sum _{x \leq n {\lt} x+y} \Lambda (n) = y + o(y) \]whenever \(x^{\theta _{\mathrm{PNT}}+ \varepsilon } \leq y \leq x^{1-\varepsilon }\).
We let \(\theta _{\mathrm{PNT-AA}}\) denote the least exponent with the following property: if \(\varepsilon {\gt} 0\) is fixed, and \(X\) is unbounded, then we have
\[ \int _X^{2X} |\sum _{x \leq n {\lt} x+y} \Lambda (n)-y|\ dx = o(Xy) \]whenever \(X^{\theta _{\mathrm{PNT-AA}}+ \varepsilon } \leq y \leq X^{1-\varepsilon }\).
We let \(\theta _{\mathrm{gap}}\) denote the least exponent such that, if \(p_n\) denotes the \(n^{\mathrm{th}}\) prime, that
\[ p_{n+1} - p_n \ll n^{\theta _{\mathrm{gap}}+o(1)} = p_n^{\theta _{\mathrm{gap}}+o(1)} \]as \(n \to \infty \).
We let \(\theta _{\mathrm{gap},2}\) denote the least exponent such that
\[ \sum _{p_n \leq x} (p_{n+1}-p_n)^2 \ll x^{\theta _{\mathrm{gap},2}+o(1)} \]as \(x \to \infty \).
We let \(\theta _{\mathrm{gap-AA}}\) denote the least exponent such that for every \(\varepsilon {\gt}0\), the intervals \([n, n^{\theta _{\mathrm{gap-AA}}+\varepsilon }]\) contain a prime for a density \(1\) set of natural numbers \(n\).
We have
and \(1 \leq \theta _{\mathrm{gap},2}\leq 1+\theta _{\mathrm{gap}}\).
These are all immediate, after noting from the prime number theorem that \(\sum _{p_n \leq x} p_{n+1} - p_n = x^{1+o(1)}\).
The Cramér random model [ 48 ] predicts
\(\theta _{\mathrm{PNT}}= 0\), and hence (by Lemma 15.2) \(\theta _{\mathrm{gap-AA}}= \theta _{\mathrm{PNT-AA}}= \theta _{\mathrm{gap}}=0\) and \(\theta _{\mathrm{gap},2}=1\).
We note that the results of Maier [ 203 ] show that there is some deviation from the prime number theorem at very small scales (of order \(\log ^{O(1)} x\)), but this does not directly affect the exponents discussed here due to the epsilons in our definitions.
A basic connection with zero density exponents is
Let
Then
and
See for instance [ 88 , § 13.2 ] .
We have \(\theta _{\mathrm{PNT}}\leq \frac{7}{12}\) and \(\theta _{\mathrm{PNT-AA}}\leq \frac{1}{6}\).
[ 88 ] We have \(\theta _{\mathrm{PNT}}\leq \frac{17}{30}\) and \(\theta _{\mathrm{PNT-AA}}\leq \frac{2}{15}\).
These are currently the best known upper bounds on \(\theta _{\mathrm{PNT}}\) and \(\theta _{\mathrm{PNT-AA}}\).
The density hypothesis implies that \(\theta _{\mathrm{PNT}}\leq 1/2\) and \(\theta _{\mathrm{PNT-AA}}= 0\).
The current unconditional best bound on \(\theta _{\mathrm{gap}}\) is
Historical bounds on \(\theta _{\mathrm{gap}}\) are summarized in the following table:
|
Reference |
Upper bound |
|
Hoheisel (1930) [ 118 ] |
\(1 - \frac{1}{33000} = 0.999\dots \) |
|
Heilbronn (1933) [ 116 ] |
\(1 - \frac{1}{250} = 0.996\) |
|
Ingham (1937) [ 137 ] |
\(\frac{5}{8} = 0.625\) |
|
Montgomery (1969) [ 217 ] |
\(\frac{3}{5} = 0.6\) |
|
Huxley (1972) [ 122 ] |
\(\frac{7}{12} = 0.5833\dots \) |
|
Iwaniec–Jutila (1979) [ 148 ] |
\(\frac{13}{23} = 0.5652\dots \) |
|
\(\frac{11}{20} = 0.55\) |
|
|
Pintz (1981) [ 233 ] |
\(\frac{17}{31} = 0.5483\dots \) |
|
Iwaniec–Pintz (1984) [ 151 ] |
\(\frac{23}{42} = 0.5476\dots \) |
|
Mozzochi (1986) [ 222 ] |
\(\frac{1051}{1920} = 0.5473\dots \) |
|
Lou–Yao (1984) [ 196 ] |
\(\frac{35}{64} = 0.5469\dots \) |
|
Lou–Yao (1992) [ 200 ] |
\(\frac{6}{11} = 0.5454\dots \) |
|
Baker-Harman (1996) [ 5 ] |
\(\frac{107}{200} = 0.535\) |
|
Baker–Harman–Pintz (2001) [ 8 ] |
\(\frac{21}{40} = 0.525\) |
|
R. Li (2025) [ 183 ] |
\(\frac{13}{25} = 0.52\) |
Bounds on \(\theta _{\mathrm{gap-AA}}\) are recorded in Table 15.2.
|
Reference |
Upper bound |
|
Selberg (1943) [ 263 ] |
\(\frac{19}{77}=0.2467\dots \) |
|
Montgomery (1971) [ 218 ] |
\(\frac{1}{5}=0.2\) |
|
Huxley (1972) [ 122 ] |
\(\frac{1}{6} = 0.1666\dots \) |
|
Harman (1982) [ 99 ] |
\(\frac{1}{10} = 0.1\) |
|
\(\frac{1}{12} = 0.0833\dots \) |
|
|
Jia (1995) [ 154 ] |
\(\frac{1}{13} = 0.0769\dots \) |
|
Lou–Yao (1985) [ 197 ] |
\(\frac{17}{227}= 0.0748\dots \) |
|
H. Li (1995) [ 178 ] |
\(\frac{2}{27} = 0.0740\dots \) |
|
\(\frac{1}{14}=0.0714\dots \) |
|
|
H. Li (1997) [ 179 ] |
\(\frac{1}{15}=0.0666\dots \) |
|
Baker–Harman–Pintz (1997) [ 7 ] |
\(\frac{1}{16} = 0.625\) |
|
Wong (1996) [ 298 ] , Jia (1996) [ 156 ] , Harman (2007) [ 101 ] |
\(\frac{1}{18} = 0.0555\dots \) |
|
Jia (1996) [ 155 ] |
\(\frac{1}{20} = 0.05\) |
|
R. Li (2024) [ 181 ] |
\(\frac{2}{43} = 0.0465\dots \) |
|
R. Li (2025) [ 185 ] |
\(\frac{1}{22} = 0.0455\dots \) |
Historical bounds on \(\theta _{\mathrm{gap},2}\) are recorded in Table 15.3.
|
Reference |
Upper bound |
|
Selberg (1943) [ 263 ] |
\(1\) (on RH) |
|
Heath-Brown (1978) [ 102 ] |
\(\frac{4}{3} = 1.3333\dots \) |
|
Heath-Brown (1979) [ 104 ] |
\(\frac{7}{6} = 1.1666\dots \) (on LH) |
|
Heath-Brown (1979) [ 104 ] |
\(\frac{1413}{1067} = 1.3242\dots \) |
|
Heath-Brown (1979) [ 105 ] |
\(\frac{23}{18} = 1.2777\dots \) |
|
Yu (1996) [ 312 ] |
\(1\) (on LH) |
|
\(\frac{5}{4} = 1.25\) |
|
|
Stadlmann (2022) [ 268 ] |
\(\frac{123}{100} = 1.23\) |
The following general bound on \(\theta _{\mathrm{gap},2}\) is known:
We have
where
and
where \(A(\sigma ), B(\sigma )\) are any upper bounds for \(\mathrm{A}(\sigma ), \mathrm{A}^*(\sigma )\) respectively.
See [ 104 , Lemma 2 ] . We remark that this lemma allows \(\sigma \) to range over \(0 \leq \sigma \leq 1\) rather than \(1/2 \leq \sigma \leq 1\), but it is easy to see that the contributions of the \(0 \leq \sigma {\lt} 1/2\) cases are dominated by the \(\sigma =1/2\) case.
compute_gap2()
This proposition can be used to recover the following bounds on \(\theta _{\mathrm{gap},2}\):
For (i), we observe that \(\| A\| _\infty =2\) and that one can take \(A(\sigma )=B(\sigma )=\varepsilon \) for any \(\sigma {\gt}1/2\) and \(\varepsilon {\gt}0\), and \(A(\sigma )=2\), \(B(\sigma )=6\) for \(\sigma =1/2\), and then the claim follows from Proposition 15.9.
For (ii), from Theorem 11.12 we may take \(A(\sigma )=2\) for \(\sigma \leq 3/4\) and \(A(\sigma )=\varepsilon \) for \(3/4 {\lt} \sigma \leq 1\) and any \(\varepsilon {\gt}0\), while from Theorem 12.5 one can take \(B(\sigma ) = 8-4\sigma \) for \(\sigma \leq 3/4\) and \(B(\sigma )=\varepsilon \) for \(3/4 {\lt} \sigma \leq 1\). The claim now follows from Proposition 15.9 and a routine calculation.
Part (iii) follows from applying Proposition 15.9 using the bounds from Theorem 12.6, together and various bounds on \(\mathrm{A}(\sigma )\); see [ 144 , Theorem 12.14 ] for details.
Two variants of \(\theta _{\mathrm{gap},2}\) are \(\theta _{\mathrm{gap},{\gt}}\) and \(\theta _{\mathrm{gap},\geq }\), defined respectively as the least exponent for which
(for any fixed \(\varepsilon {\gt}0\) for unbounded \(x \geq 1\)) and
(for unbounded \(x \geq 1\)). The trivial bounds are
One has \(\theta _{\mathrm{gap},{\gt}}\le \theta _{\mathrm{gap},\geq }\). If \(\theta _{\mathrm{gap}}{\lt} 1/2\), then \(\theta _{\mathrm{gap},{\gt}}= -\infty \). In general, we have
and \(\theta _{\mathrm{gap},{\gt}}\leq 1\). Also \(\theta _{\mathrm{gap},{\gt}}\leq \theta _{\mathrm{gap},2}- 1/2\).
The proofs are routine and are omitted. Historical bounds on \(\theta _{\mathrm{gap},{\gt}}\) are recorded in Table 15.4.
|
Reference |
Upper bound on \(\theta _{\mathrm{gap},{\gt}}\) |
Upper bound on \(\theta _{\mathrm{gap},\geq }\) |
|
Selberg (1943) [ 263 ] |
\(\frac{1}{2}=0.5\) (on RH) |
|
|
Wolke (1975) [ 297 ] |
\(\frac{29}{30} = 0.966\dots \) |
|
|
Cook (1979) [ 44 ] |
\(\frac{85}{98} = 0.8673\dots \) |
|
|
Huxley (1980) [ 126 ] |
\(\frac{1759}{2134} = 0.8242\dots \) |
|
|
Huxley (1980) [ 126 ] |
\(\frac{3}{4} = 0.75\) (on LH) |
|
|
Ivíc (1979) [ 140 ] |
\(\frac{215}{266} = 0.8082\dots \) |
|
|
Heath-Brown (1979) [ 105 ] |
\(\frac{3}{4} = 0.75\) |
|
|
Heath-Brown (1979) [ 104 ] |
\(\frac{5}{8} = 0.625\) |
|
|
Peck (1998) [ 231 ] |
\(\frac{25}{36} = 0.6944\ldots \) |
|
|
Matomäki (2007) [ 205 ] |
\(\frac{2}{3} = 0.6666\ldots \) |
|
|
Heath-Brown (2020) [ 114 ] |
\(\frac{3}{5} = 0.6\) |
|
|
Järviniemi (2022) [ 152 ] |
\(\frac{57}{100} = 0.57\) |
For any \(0 {\lt} \theta {\lt} 1\), let \(\mu _{\mathrm{PNT}}(\theta )\) denote the least exponent \(\mu \) such that for all unbounded \(X\), one has \(\sum _{x \leq n {\lt} x + x^\theta } \Lambda (n) = (1+o(1)) x^\theta \) for all \(x \in [X,2X]\) outside of an exceptional set of measure \(O(X^{\mu +o(1)})\). Thus for instance \(\mu _{\mathrm{PNT}}(\theta )=-\infty \) for \(\theta {\gt} \theta _{\mathrm{PNT}}\) (and \(\mu _{\mathrm{PNT}}(\theta ) \geq 0\) for \(\theta {\lt} \theta _{\mathrm{PNT}}\)), and \(\mu _{\mathrm{PNT}}(\theta ) {\lt} 1\) implies \(\theta \geq \theta _{\mathrm{PNT-AA}}\). The quantity \(\mu _{\mathrm{PNT}}(\theta )\) is clearly non-decreasing in \(\theta \).
The following bounds are known:
[ 12 , Theorem 2(i) ] For sufficiently small \(\Delta {\gt}0\), we have \(\mu _{\mathrm{PNT}}(1/6 + \Delta ) \leq 1 - c\Delta \) and \(\mu _{\mathrm{PNT}}(7/12-\Delta ) \leq \frac{5}{8} + \frac{7}{4}\Delta + O(\Delta ^2)\).
[ 12 , Theorem 2(ii) ] Assuming RH, we have \(\mu _{\mathrm{PNT}}(\theta ) \leq 1-\theta \) for \(0 {\lt} \theta \leq 1/2\).
[ 11 , Lemma 1 ] We have
\[ \mu _{\mathrm{PNT}}(\theta ) \leq \begin{cases} \frac{3(1-\theta )}{2} & \frac{1}{2} {\lt} \theta \leq \frac{11}{21} \\ \frac{47-42\theta }{35}& \frac{11}{21} {\lt} \theta \leq \frac{23}{42} \\ \frac{36\theta ^2-96\theta +55}{39-36\theta } & \frac{23}{42} {\lt} \theta \leq \frac{7}{12} \\ \end{cases} \]Some further bounds were claimed in the region \(1/6 {\lt} \theta \leq 1/2\), but unfortunately the arguments provided are incomplete (the claim (13) of that paper is not justified for \(\theta \leq 1/2\)).
[ 78 ] For any \(0 {\lt} \theta {\lt} 1\), one has
\[ \mu _{\mathrm{PNT}}(\theta ) \leq \inf _{\varepsilon {\gt}0} \sup _{\stackrel{0 \leq \sigma {\lt} 1}{\mathrm{A}(\sigma ) \geq \frac{1}{1-\theta }-\varepsilon }} \min (\mu _{2,\sigma }(\theta ), \mu _{4,\sigma }(\theta )) \]where
\[ \mu _{2,\sigma }(\theta ) := (1-\theta )(1-\sigma ) \mathrm{A}(\sigma ) + 2\sigma - 1 \]and
\[ \mu _{4,\sigma }(\theta ) := (1-\theta )(1-\sigma ) \mathrm{A}^*(\sigma ) + 4\sigma -3. \]
[(v)] [ 114 , Theorem 2 ] \(\mu _{\mathrm{PNT}}(1/2) \leq 3/5\).
prime_excep()
In 2004, under the assumption of the existence of exceptional Dirichlet characters, Friedlander and Iwaniec [ 77 ] proved the following result:
[ 77 ] Let \(\chi =\chi _{D}\) denotes the real primitive character of conductor D, \(x \geqslant D^{r}\) with \(r=18290\). Then we have
Moreover, if we have
then there is always a prime number in the interval \(\left[x - x^{\frac{39}{79}}, x \right]\) for any \(D^{r} \leqslant x \leqslant \exp \left(L(1, \chi )^{- \frac{1}{r^r + 1} } \right)\).
Note that \(\frac{39}{79} = 0.4936 \dots \). In 2024, Li [ 180 ] improved the exponent \(\frac{39}{79}\) to \(0.4923\) with \(r=433433\).
15.1 Extremal values of prime gaps
Consider now the problem of determining upper bounds on
as well as lower bounds on
From the prime number theorem one expects \(p_{n + 1} - p_n\) to be of size \(\log p_n\) on average, so that
One has
However, \(p_{n + 1} - p_n\) can be sometimes be much smaller or much larger than its average size. The following is a classical conjecture regarding small prime gaps.
One has
Since all sufficiently large primes are odd, the twin prime conjecture states that prime gaps achieve the smallest possible size, infinitely often. In the other direction, it is conjectured that
Note that by Theorem 15.8 it is known that \(G(X) \ll X^{0.52}\).
The current best known result concerning 6 is
Sharper conditional bounds are also known.
Assuming the Elliott-Halberstam conjecture (EH), one has
Assuming the Generalized Elliott-Halberstam conjecture (GEH), one has
Historical progress towards this problem is recorded in Table 15.5.
|
Reference |
Unconditional result |
Assuming EH |
|
Goldston–Pintz–Yıldırım (2009) [ 82 ] |
\(\displaystyle \liminf _{n\to \infty }\frac{p_{n + 1} - p_n}{\log p_n} = 0\) |
\(H_1 \le 16\) |
|
Goldston–Pintz–Yıldırım (2010) [ 83 ] |
\(\displaystyle \liminf _{n\to \infty }\frac{p_{n + 1} - p_n}{(\log p_n)^{1/2}(\log \log p_n)^2} {\lt} \infty \) |
|
|
Pintz (2013) [ 235 ] |
\(\displaystyle \liminf _{n\to \infty }\frac{p_{n + 1} - p_n}{(\log p_n)^{3/7}(\log \log p_n)^{4/7}} {\lt} \infty \) |
|
|
Zhang (2014) [ 315 ] |
\(H_1 {\lt} 7\cdot 10^7\) |
|
|
Polymath 8a (2014) [ 30 ] |
\(H_1 \le 4680\) |
|
|
Maynard (2015) [ 209 ] |
\(H_1 \le 600\) |
\(H_1 \le 12\) |
|
Polymath 8b (2015) [ 240 ] |
\(H_1 \le 246\) |
The current best known lower bound on \(G(X)\) is
For unbounded \(X\), one has
|
Reference |
Lower bound on \(G(X)\) (for \(X\) sufficiently large) |
|
Westzynthius (1931) [ 295 ] |
\(\displaystyle G(X) \gg \log X \frac{\log \log \log X}{\log \log \log \log X}\) |
|
Erdős (1935) [ 65 ] |
\(\displaystyle G(X) \gg \log X \frac{\log \log X}{(\log \log \log X)^2}\) |
|
Rankin (1938) [ 244 ] |
\(\displaystyle G(X) {\gt} (c_0 + o(1))\log X \frac{\log \log X \log \log \log \log X}{(\log \log \log X)^2}\) with \(c_0 = 1/3\) |
|
Schönhage (1963) [ 262 ] |
\(c_0 = \dfrac {1}{2}e^\gamma \) |
|
Rankin (1963) [ 246 ] |
\(c_0 = e^\gamma \) |
|
Maier–Pomerance (1990) [ 204 ] |
\(c_0 = 1.31256 e^\gamma \) |
|
Pintz (1997) [ 234 ] |
\(c_0 = 2e^\gamma \) |
|
Ford–Green–Konyagin–Tao (2016) [ 69 ] , Maynard (2016) [ 210 ] |
\(\displaystyle G(X) \gg f(X)\log X \frac{\log \log X \log \log \log \log X}{(\log \log \log X)^2}\) for some \(f(X) \to \infty \) |
|
Ford–Green–Konyagin–Maynard–Tao (2017) [ 68 ] |
\(\displaystyle G(X) \gg \frac{\log X \log \log X \log \log \log \log X}{\log \log \log X}\) |