Exponent for bounded gaps between many primes
Description of constant
Let $p_n$ denote the $n$-th prime and, for $m \ge 1$, write
\[H_m := \liminf_{n \to \infty} (p_{n+m} - p_n)\]for the least limit point of the gaps between primes $m$ apart. Maynard [May15] proved that $H_m$ is finite for every $m$, with $H_m \ll m^3 e^{4m}$; the growth rate in $m$ is exponential in all bounds known, and the exponent for bounded gaps between many primes is
\[C_{88b} := \limsup_{m \to \infty} \frac{\log H_m}{m},\]equivalently the least $c$ for which $H_m \ll_\varepsilon \exp((c+\varepsilon)m)$ holds for every $\varepsilon > 0$.
The case $m = 1$ is recorded separately as $C_{88a}$; note that $C_{88b}$ is insensitive to any finite number of $H_m$, so the two are genuinely independent quantities.
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $4$ | [May15] | $H_m \ll m^3 e^{4m}$, the first finiteness result for $m \ge 2$, from the multidimensional Selberg sieve. |
| $4 - \frac{28}{157} = 3.821656\dots$ | [Pol14b] | $H_m \ll m e^{(4-28/157)m}$. Polymath8b, quoted in this form by [Sta26]. |
| $3.815$ | [BI17] | Baker–Irving. |
| $3.8075$ | [Sta25] | Stadlmann: $H_m = O(\exp(3.8075\,m))$, deduced from an exponent of distribution $\tfrac12 + \tfrac1{40}$ for the primes in arithmetic progressions to smooth moduli, improving Polymath’s $\tfrac12 + \tfrac7{300}$. The new ingredient is a modification of the $q$-van der Corput process. |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $0$ | Trivial | No positive lower bound is known, and none is expected: see below. |
Additional comments and links
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The expected value is $0$. The prime $k$-tuples conjecture predicts $H_m \asymp m \log m$, which is sub-exponential, so conjecturally $C_{88b} = 0$. The interest is therefore entirely in the upper bound, as for $C_{62a}$ and other exponents in this repository whose conjectural value is $0$.
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Conditional bound. Assuming the Elliott–Halberstam conjecture, Maynard obtained $H_m \ll m^3 e^{2m}$ [May15], as reported in the abstract of [Pol14b], giving $C_{88b} \le 2$ under EH.
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Caution: this does not improve $C_{66}$. The exponents of distribution appearing here — Polymath’s $\tfrac12 + \tfrac7{300}$ and Stadlmann’s $\tfrac12 + \tfrac1{40} = 0.525$ — are established only for smooth (friable) moduli, and with well-factorable weights. $C_{66}$ as defined on its own page is the supremum of admissible levels for the full sum over all moduli $q \le Q$, for which $\tfrac12$ (Bombieri–Vinogradov) remains the record. The two must not be conflated: $\tfrac12 + \tfrac1{40}$ is not a lower bound for $C_{66}$.
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Relation to the other entries. $C_{88a}$ is $H_1$ itself. Both this constant and $C_{88a}$ improved as a direct consequence of the same equidistribution input, [Sta25], which is what [Sta26] then combined with Bombieri–Vinogradov to reach $H_1 \le 240$.
References
- [Pol14b] D. H. J. Polymath. Variants of the Selberg sieve, and bounded intervals containing many primes. Research in the Mathematical Sciences 1 (2014), Art. 12. DOI: 10.1186/s40687-014-0012-7. Erratum: ibid. 2 (2015), Art. 15, DOI: 10.1186/s40687-015-0033-x. arXiv:1407.4897.
- [May15] Maynard, James. Small gaps between primes. Annals of Mathematics 181 (2015), no. 1, 383–413. DOI: 10.4007/annals.2015.181.1.7.
- [BI17] Baker, Roger C.; Irving, Andrew J. Bounded intervals containing many primes. Mathematische Zeitschrift 286 (2017), no. 3–4, 821–841. DOI: 10.1007/s00209-016-1786-7.
- [Sta25] Stadlmann, Julia. On primes in arithmetic progressions and bounded gaps between many primes. Advances in Mathematics 468 (2025), Paper No. 110190. DOI: 10.1016/j.aim.2025.110190. arXiv:2309.00425.
- [Sta26] Stadlmann, Julia. Bounded gaps between primes. Preprint, 31 August 2026. arXiv:2608.31126.
Contribution notes
Prepared with assistance from Claude Opus 5. The chain $4 \to 4-\tfrac{28}{157} \to 3.815 \to 3.8075$ is as set out in the introduction of [Sta26]; the value $3.8075$ and the exponent of distribution $\tfrac12+\tfrac1{40}$ are quoted from the abstract of [Sta25]. Bibliographic details for all five references were confirmed through Crossref, which also showed that [Sta25] has appeared in Advances in Mathematics rather than being merely accepted. No bound was independently verified, and the Polymath and Baker–Irving constants were not checked against their primary sources.