Large prime gaps occur with positive frequency

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Family 026 · Number theory
Importance 3/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

The primes become less frequent as numbers grow. Near a large number $x$, their average spacing is about $\log x$, where $\log$ denotes the natural logarithm. Some consecutive primes are much closer together, while others are much farther apart. How frequently does a gap exceed, say, ten times the local average?

Write $p_n$ for the $n$th prime. The manuscript claims that, for every fixed positive number $C$, a positive proportion of the gaps satisfy

$$p_{n+1}-p_n>C\log p_n.$$

The proportion may depend on $C$ and become very small as $C$ grows. Nevertheless, it stays bounded away from zero as we inspect longer initial segments of the prime sequence.

There are two different questions here. One asks whether an exceptionally large gap exists somewhere. The other asks whether large gaps constitute a positive fraction of all gaps. A single enormous desert between primes answers the first question but contributes only one gap to the second. The new claim concerns the second question, with each prime gap counted once.

2. Background and essential definitions

The prime-counting function $\pi(x)$ counts primes at most $x$. The prime number theorem says

$$\pi(x)\sim\frac{x}{\log x},$$

meaning that the ratio of the two sides tends to one. Consequently, the average gap on a large scale is logarithmic. Define the normalized gap by

$$g_n=\frac{p_{n+1}-p_n}{\log p_n}.$$

Dividing by the local average makes gaps at different heights comparable.

A set $A$ of positive integers has positive lower asymptotic density if

$$\liminf_{N\to\infty}\frac{|A\cap\{1,\ldots,N\}|}{N}>0.$$

The vertical bars count elements. This condition says that some fixed positive fraction works for every sufficiently large cutoff. It is stronger than obtaining a positive fraction along a selected sequence of cutoffs. It does not assert that the fraction has a limit.

Two ways to sample empty space

One can choose a prime and examine the gap following it. Alternatively, one can choose an integer $m$ and ask whether $(m,m+h]$ contains a prime. These experiments weight the same prime desert differently. A gap of length $D$ may contain roughly $D-h$ starting points of empty intervals of length $h$, but it is still only one consecutive prime gap.

This distinction is central. A statement that many integer-start intervals are empty does not automatically imply that many prime-start gaps are large. The latter needs control of how many interval starts can correspond to the same gap.

A probabilistic model in which primes appear independently with probability roughly $1/\log x$ suggests an exponential tail for normalized gaps. Such a model is useful intuition, not a theorem about the primes. The manuscript establishes positivity of every fixed tail, without identifying that tail with the model’s value.

3. Development history and previous partial results

Westzynthius proved in 1931 that normalized prime gaps are unbounded. Erdős and Rankin subsequently improved quantitative lower bounds for exceptionally large gaps. More recently, Ford, Green, Konyagin and Tao, and independently Maynard, strengthened these constructions; their joint work gave further improvements. These are developments in the size of the largest gaps, rather than a complete description of how often large gaps occur.

The frequency problem needs different care. Bazzanella, Languasco and Zaccagnini’s Prime numbers in logarithmic intervals explicitly distinguishes integer-start intervals from prime-start gaps. As the new manuscript records, their Theorem 5 gives positive prime-start proportions for fixed thresholds below approximately $0.579038$ times the average spacing. Other conclusions in their work concern integer-start intervals and therefore cannot simply be read as prime-gap proportions above the average.

There are also much more detailed conditional predictions. Gallagher obtained Poisson statistics for prime counts in logarithmic intervals from a sufficiently uniform Hardy–Littlewood prime-tuple hypothesis. Roughly, that hypothesis estimates how often several specified integer shifts are simultaneously prime. Uniformity matters because the shifts and the intervals being counted vary. More recent conditional work extends the parameter ranges, but still uses strong assumptions about prime tuples.

The manuscript’s analytic ancestry includes the Bombieri–Vinogradov theorem, which controls prime distribution in arithmetic progressions after averaging over the moduli. It also builds on Goldston–Yıldırım’s correlations of divisor sums and Maynard’s smooth multidimensional sieve weights. A sieve weight is a numerical function designed to emphasize integers with chosen divisibility properties. Its usefulness comes from having averages that can be evaluated even when the corresponding prime pattern cannot be counted directly.

The proposed advance combines these ideas to handle arbitrary fixed $C$, unconditionally as stated, in the prime-start counting measure. It also addresses a 1962 question of Erdős and Prachar concerning the sequence $p_n/n$.

4. What the new manuscripts claim

There is one manuscript in this family. Its main theorem states that for every fixed real $C>0$, there are constants $c(C)>0$ and $N_0(C)$ such that

$$\#\{1\le n\le N:p_{n+1}-p_n>C\log p_n\} \ge c(C)N \qquad(N\ge N_0(C)).$$

Here $\#$ means the number of indices satisfying the condition. Both constants may depend on $C$. In particular, the threshold is fixed before $N$ tends to infinity.

The theorem gives ordinary lower density among prime indices. It requires neither an average over cutoffs nor a logarithmic weighting of indices. It is also not merely an assertion of infinitely many examples.

An application to the ratio of a prime to its index

The manuscript deduces that

$$\left\{n:\frac{p_{n+1}}{n+1}>\frac{p_n}{n}\right\}$$

has positive lower density. To understand the connection, multiply by $n(n+1)$. The displayed inequality is equivalent to

$$p_{n+1}-p_n>\frac{p_n}{n}.$$

The prime number theorem gives $p_n/n\sim\log p_n$. Thus a gap exceeding twice $\log p_n$ eventually suffices. The main theorem with $C=2$ supplies a positive proportion of such indices. This is the stated affirmative answer to the Erdős–Prachar question.

Although $p_n/n$ has an overall tendency to grow, that broad asymptotic behavior alone would not imply a positive proportion of individual increases. The corollary concerns the local steps of this sequence.

5. Why the advance matters

Importance: 3/5, provisionally. The claim is a substantial advance in the distribution of prime gaps: it reaches every fixed multiple of the average spacing with a frequency conclusion, and resolves a specific longstanding question. The rating also reflects its limits. It does not determine the full gap distribution or settle the most precise conjectures about extreme gaps.

The result would sharpen the qualitative picture of prime irregularity. Arbitrarily large normalized gaps would be accompanied by a stronger statement: for any fixed level, exceeding that level is a recurring phenomenon with positive lower frequency.

The method also addresses a general counting problem. A weighted or geometric argument can count many witnesses while repeatedly witnessing the same object. Here the manuscript introduces a nearby prime as an anchor, giving a uniform bound on that repetition. That counting step makes the sieve construction relevant to the desired statistic.

6. The main idea

Put a prime immediately before an empty block

Choose a fixed $\lambda>C$, and let $h=\lfloor\lambda\log X\rfloor$. For integers $m$ between $X$ and $2X$, consider two adjacent intervals:

$$J_m=(m,m+h],\qquad I_m=(m+h,m+2h].$$

The target is a prime in $J_m$ and no primes in $I_m$. If $p$ is the last prime in the first interval, its successor lies beyond the second interval, so the gap after $p$ exceeds $h$.

Crucially, a fixed prime $p$ can be selected by at most $h$ different starting integers $m$, since $m<p\le m+h$. This bound does not grow with the length of the actual prime gap. Therefore $\delta X$ successful interval starts yield at least $\delta X/h$ distinct prime gaps, where $\delta>0$. Since $h$ is logarithmic, this is the same order of magnitude as the number of primes in the region.

Build a weight with unequal behavior in adjacent intervals

The manuscript constructs nonnegative weights $W_X(m)$. Under these weights, the first interval retains substantial prime mass, while the expected prime count in the second can be made arbitrarily small. The weights are normalized to have average tending to one.

Two second-moment controls serve different purposes. One prevents the first interval’s detected prime mass from concentrating on too little weighted mass. The other bounds the average of $W_X(m)^2$, preventing the useful weighted mass from concentrating on too few ordinary integers. The Cauchy–Schwarz inequality then converts the weighted statements into a positive ordinary proportion of successful starts.

The order of choices matters: the constant controlling prime detection in the first interval is independent of how strongly primes are suppressed in the second. The bound for the weights’ own second moment can depend on that suppression. Everything is fixed before $X$ grows.

Obtain suppression by cancellation inside a square

A nonnegative weight cannot cancel directly. Instead, the paper forms a signed sum of smooth divisor sums and then squares it. Each divisor sum corresponds to a set of shifts in the second interval. Different set sizes act like different dimensions of the construction.

The chosen test functions make distinct shift sets orthogonal in an unmarked average: their cross terms disappear in the limiting formula. If one marks a particular shift as prime, a support restriction forces its divisor coordinate to be one. Removing that coordinate couples terms whose dimensions differ by one. Alternating coefficients make those neighboring contributions nearly cancel.

Thus the square retains positive total mass while its interaction with primes in the second interval is small. Smooth divisor correlations, averaged prime distribution and averages of the prime-tuple singular series evaluate the required moments. The singular series records compatibility with divisibility conditions at each prime. The manuscript keeps all dimensions and test functions fixed, proving the uniform estimates needed to sum over the shifts.

7. Scope, limitations, and open questions

The theorem gives a positive lower bound depending on each fixed $C$. It does not supply a sharp expression for that bound, establish a limiting gap distribution, or give the exponential tail suggested by an independent random model.

It does not permit replacing $C$ by an arbitrary function growing with $N$. Results on the largest prime gap investigate a different regime and remain relevant independently.

The distinction between lower density and density also remains: the proportions are bounded away from zero eventually, but need not be shown to converge. Likewise, the corollary does not identify the exact proportion of indices where $p_n/n$ increases.

All conclusions here report the manuscript’s claims. The central conceptual gain is the conversion from empty intervals to distinct consecutive gaps with controlled multiplicity; the resulting assertion is stronger than an empty-interval frequency statement alone.

8. Sources and further reading

The source manuscript below contains the precise weight proposition, its moment identities and the passage to lower density in prime indices.

For the history of extreme gaps, see Ford, Green, Konyagin, Maynard and Tao, Long gaps between primes. For the divisor-sum method, see Goldston and Yıldırım, Small gaps between primes I, and Maynard, Small gaps between primes. These primary sources explain why related sieve technology can address quite different statistics.

The bibliography of M1 identifies the original Westzynthius, Erdős–Rankin, Erdős–Prachar, Gallagher and Bazzanella–Languasco–Zaccagnini papers. Its introduction also separates their counting measures and hypotheses, which is essential when comparing the results.

Source manuscripts

• M1. OpenAI, Positive lower density of large prime gaps (September 25, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.