13 Applications to the primes

Recall that $Λ$ is the von Mangoldt function, and that the prime number theorem asserts that

\sum_{n \leq x} Λ (n) = x + o (x)

for unbounded $x$ . If $p_{n}$ denotes the $n^{th}$ prime, the prime number theorem is also equivalent to

p_{n} = (1 + o (1)) n \log n

for unbounded $n$ .

We now consider local versions of the prime number theorem.

Definition 13.1 Prime number theorem in short interval exponents

We let $θ_{PNT}$ denote the least exponent with the following property: if $ε > 0$ is fixed, and $x$ is unbounded, then

$\sum_{x \leq n < x + y} Λ (n) = y + o (y)$

whenever $x^{θ_{PNT} + ε} \leq y \leq x^{1 - ε}$ .
We let $θ_{PNT - AA}$ denote the least exponent with the following property: if $ε > 0$ is fixed, and $X$ is unbounded, then we have

$\int_{X}^{2 X} | \sum_{x \leq n < x + y} Λ (n) - y | d x = o (X y)$

whenever $X^{θ_{PNT - AA} + ε} \leq y \leq X^{1 - ε}$ .
We let $θ_{gap}$ denote the least exponent such that, if $p_{n}$ denotes the $n^{th}$ prime, that

$p_{n + 1} - p_{n} ≪ n^{θ_{gap} + o (1)} = p_{n}^{θ_{gap} + o (1)}$

as $n \to \infty$ .
We let $θ_{gap, 2}$ denote the least exponent such that

$\sum_{p_{n} \leq x} (p_{n + 1} - p_{n})^{2} ≪ x^{θ_{gap, 2} + o (1)}$

as $x \to \infty$ .
We let $θ_{gap - AA}$ denote the least exponent such that for every $ε > 0$ , the intervals $[n, n^{θ_{gap - AA} + ε}]$ contain a prime for a density $1$ set of natural numbers $n$ .

Lemma 13.2 Trivial bounds

We have

0 \leq θ_{gap - AA} \leq θ_{PNT - AA}, θ_{gap} \leq θ_{PNT} \leq 1

and $1 \leq θ_{gap, 2} \leq 1 + θ_{gap}$ .

Proof ▶

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 [ 38 ] predicts

Heuristic 13.3 Prime gap conjecture

$θ_{PNT} = 0$ , and hence (by Lemma 13.2) $θ_{gap - AA} = θ_{PNT - AA} = θ_{gap} = 0$ and $θ_{gap, 2} = 1$ .

We note that the results of Maier [ 160 ] 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

Proposition 13.4 Zero density theorems and prime gaps

Let

‖ A ‖_{\infty} := sup_{1 / 2 \leq σ \leq 1} A (σ) .

Then

θ_{PNT} \leq 1 - \frac{1}{‖ A ‖_{\infty}}

and

θ_{PNT - AA} \leq 1 - \frac{2}{‖ A ‖_{\infty}} .

Proof ▶

Corollary 13.5 Ingham-Huxley bound

We have $θ_{PNT} \leq \frac{7}{12}$ and $θ_{PNT - AA} \leq \frac{1}{6}$ .

Proof ▶

From Theorem 11.14 and Theorem 11.15 one as $‖ A ‖_{\infty} \leq 12 / 5$ , and the claim now follows from Proposition 13.4.

Corollary 13.6 Ingham-Guth-Maynard bound

[ 61 ] We have $θ_{PNT} \leq \frac{17}{30}$ and $θ_{PNT - AA} \leq \frac{2}{15}$ .

These are currently the best known upper bounds on $θ_{PNT}$ and $θ_{PNT - AA}$ .

Proof ▶

From Theorem 11.14 and Theorem 11.16 one as $‖ A ‖_{\infty} \leq 30 / 13$ , and the claim now follows from Proposition 13.4.

Corollary 13.7

The density hypothesis implies that $θ_{PNT} \leq 1 / 2$ and $θ_{PNT - AA} = 0$ .

The current unconditional best bound on $θ_{gap}$ is

Theorem 13.8

[ 158 ] We have $θ_{gap} \leq 259 / 500 = 0.518$ .

Historical bounds on $θ_{gap}$ are summarized in the following table:

Table 13.1 Historical upper bounds on

θ_{gap}

Reference	Upper bound
Hoheisel (1930) [ 88 ]	$1 - \frac{1}{33000} = 0.999 \dots$
Heilbronn (1933) [ 86 ]	$1 - \frac{1}{250} = 0.996$
Ingham (1937) [ 103 ]	$\frac{5}{8} = 0.625$
Montgomery (1969) [ 168 ]	$\frac{3}{5} = 0.6$
Huxley (1972) [ 89 ]	$\frac{7}{12} = 0.5833 \dots$
Iwaniec–Jutila (1979) [ 116 ]	$\frac{13}{23} = 0.5652 \dots$
Heath-Brown–Iwaniec (1979) [ 84 ] , Lou–Yao (1993) [ 157 ]	$\frac{11}{20} = 0.55$
Pintz (1981) [ 183 ]	$\frac{17}{31} = 0.5483 \dots$
Iwaniec–Pintz (1984) [ 117 ]	$\frac{23}{42} = 0.5476 \dots$
Mozzochi (1986) [ 173 ]	$\frac{1051}{1920} = 0.5473 \dots$
Lou–Yao (1992) [ 156 ]	$\frac{6}{11} = 0.5454 \dots$
Baker-Harman (1996) [ 5 ]	$\frac{107}{200} = 0.535$
Baker–Harman–Pintz (2001) [ 7 ]	$\frac{21}{40} = 0.525$
R. Li (2025) [ 145 ]	$\frac{13}{25} = 0.52$
Lu–Yuan (2025) [ 158 ]	$\frac{259}{500} = 0.518$

Bounds on $θ_{gap - AA}$ are recorded in Table 13.2.

Table 13.2 Historical upper bounds on

θ_{gap - AA}

Reference	Upper bound
Selberg (1943) [ 205 ]	$\frac{19}{77} = 0.2467 \dots$
Montgomery (1971) [ 170 ]	$\frac{1}{5} = 0.2$
Huxley (1972) [ 89 ]	$\frac{1}{6} = 0.1666 \dots$
Harman (1982) [ 72 ]	$\frac{1}{10} = 0.1$
Harman (1983) [ 73 ] , Heath-Brown (1983) [ 81 ]	$\frac{1}{12} = 0.0833 \dots$
Jia (1995) [ 119 ]	$\frac{1}{13} = 0.0769 \dots$
Lou–Yao (1985) [ 153 ]	$\frac{17}{227} = 0.0748 \dots$
H. Li (1995) [ 142 ]	$\frac{2}{27} = 0.0740 \dots$
Jia (1995) [ 121 ] , Watt (1995) [ 228 ]	$\frac{1}{14} = 0.0714 \dots$
H. Li (1997) [ 143 ]	$\frac{1}{15} = 0.0666 \dots$
Baker–Harman–Pintz (1997) [ 6 ]	$\frac{1}{16} = 0.625$
Wong (1996) [ 231 ] , Jia (1996) [ 120 ] , Harman (2007) [ 71 ]	$\frac{1}{18} = 0.0555 \dots$
Jia (1996) [ 122 ]	$\frac{1}{20} = 0.05$
R. Li (2024) [ 144 ]	$\frac{2}{43} = 0.0465 \dots$

Historical bounds on $θ_{gap, 2}$ are recorded in Table 13.3.

Table 13.3 Historical upper bounds on

θ_{gap, 2}

Reference	Upper bound
Selberg (1943) [ 205 ]	$1$ (on RH)
Heath-Brown (1978) [ 74 ]	$\frac{4}{3} = 1.3333 \dots$
Heath-Brown (1979) [ 76 ]	$\frac{7}{6} = 1.1666 \dots$ (on LH)
Heath-Brown (1979) [ 76 ]	$\frac{1413}{1067} = 1.3242 \dots$
Yu (1996) [ 238 ]	$1$ (on LH)
Heath-Brown (1979) [ 77 ]	$\frac{23}{18} = 1.2777 \dots$
Peck (1996) [ 180 ] , Maynard (2012) [ 163 ]	$\frac{5}{4} = 1.25$
Stadlmann (2022) [ 209 ]	$\frac{123}{100} = 1.23$

The following general bound on $θ_{gap, 2}$ is known:

Proposition 13.9

We have

θ_{gap, 2} \leq max (2 - \frac{2}{‖ A ‖_{\infty}}, sup_{1 / 2 \leq σ \leq 1} max (α (σ), β (σ)))

where

α (σ) := 4 σ - 2 + 2 \frac{B (σ) (1 - σ) - 1}{B (σ) - A (σ)}

and

β (σ) := 4 σ - 2 + \frac{B (σ) (1 - σ) - 1}{A (σ)}

where $A (σ), B (σ)$ are any upper bounds for $A (σ), A^{*} (σ)$ respectively.

Proof ▶

See [ 76 , Lemma 2 ] . We remark that this lemma allows $σ$ to range over $0 \leq σ \leq 1$ rather than $1 / 2 \leq σ \leq 1$ , but it is easy to see that the contributions of the $0 \leq σ < 1 / 2$ cases are dominated by the $σ = 1 / 2$ case.

This proposition can be used to recover the following bounds on $θ_{gap, 2}$ :

Corollary 13.10

Assuming the Riemann hypothesis, $θ_{gap, 2} = 1$ . (Selberg, 1943 [ 205 ] )
Assuming the Lindelof hypothesis, $θ_{gap, 2} \leq 7 / 6$ . (Heath-Brown, 1979 [ 76 ] )
Unconditionally, $θ_{gap, 2} \leq 23 / 18$ . (Heath-Brown, 1979 [ 77 ] ).

Proof ▶

For (i), we observe that $‖ A ‖_{\infty} = 2$ and that one can take $A (σ) = B (σ) = ε$ for any $σ > 1 / 2$ and $ε > 0$ , and $A (σ) = 2$ , $B (σ) = 6$ for $σ = 1 / 2$ , and then the claim follows from Proposition 13.9.

For (ii), from Theorem 11.12 we may take $A (σ) = 2$ for $σ \leq 3 / 4$ and $A (σ) = ε$ for $3 / 4 < σ \leq 1$ and any $ε > 0$ , while from Theorem 12.5 one can take $B (σ) = 8 - 4 σ$ for $σ \leq 3 / 4$ and $B (σ) = ε$ for $3 / 4 < σ \leq 1$ . The claim now follows from Proposition 13.9 and a routine calculation.

Part (iii) follows from applying Proposition 13.9 using the bounds from Theorem 12.6, together and various bounds on $A (σ)$ ; see [ 110 , Theorem 12.14 ] for details. □

Two variants of $θ_{gap, 2}$ are $θ_{gap, >}$ and $θ_{gap, \geq}$ , defined respectively as the least exponent for which

\sum_{p_{n} \leq x : p_{n + 1} - p_{n} \geq x^{1 / 2 + ε}} (p_{n + 1} - p_{n}) ≪ x^{θ_{gap, >} + o (1)}

(for any fixed $ε > 0$ for unbounded $x \geq 1$ ) and

\sum_{p_{n} \leq x : p_{n + 1} - p_{n} \geq x^{1 / 2}} (p_{n + 1} - p_{n}) ≪ x^{θ_{gap, \geq} + o (1)}

(for unbounded $x \geq 1$ ). The trivial bounds are

Proposition 13.11 Trivial bounds on large gaps

One has $θ_{gap, >} \leq θ_{gap, \geq}$ . If $θ_{gap} < 1 / 2$ , then $θ_{gap, >} = - \infty$ . In general, we have

max (1 / 2, θ_{gap}) \leq max (1 / 2, θ_{gap, >})

and $θ_{gap, >} \leq 1$ . Also $θ_{gap, >} \leq θ_{gap, 2} - 1 / 2$ .

The proofs are routine and are omitted. Historical bounds on $θ_{gap, >}$ are recorded in Table 13.4.

Table 13.4 Historical upper bounds on

θ_{gap, >}

and

θ_{gap, \geq}

Reference	Upper bound on $θ_{gap, >}$	Upper bound on $θ_{gap, \geq}$
Selberg (1943) [ 205 ]	$\frac{1}{2} = 0.5$ (on RH)
Wolke (1975) [ 230 ]		$\frac{29}{30} = 0.966 \dots$
Cook (1979) [ 34 ]	$\frac{85}{98} = 0.8673 \dots$
Huxley (1980) [ 90 ]	$\frac{1759}{2134} = 0.8242 \dots$
Huxley (1980) [ 90 ]	$\frac{3}{4} = 0.75$ (on LH)
Ivíc (1979) [ 108 ]	$\frac{215}{266} = 0.8082 \dots$
Heath-Brown (1979) [ 77 ]		$\frac{3}{4} = 0.75$
Heath-Brown (1979) [ 76 ]	$\frac{5}{8} = 0.625$
Peck (1998) [ 181 ]		$\frac{25}{36} = 0.6944 \dots$
Matomäki (2007) [ 161 ]		$\frac{2}{3} = 0.6666 \dots$
Heath-Brown (2020) [ 85 ]		$\frac{3}{5} = 0.6$
Järviniemi (2022) [ 118 ]		$\frac{57}{100} = 0.57$