Primes in short intervals: Heuristics and calculations

../

conjecture_p12: The paper's summary of its conjectures on M(x,y) and m(x,y): M(x,y) = S(y) for y up to (1 - epsilon) log x, M(x,y) ~ L(x,y) for log x <= y = o((log x)^2), the u_-(c_- t) and u_+(c_+ t) asymptotics for y = t(log x)^2, and sigma_-(A) and sigma_+(A) for y = (log x)^A with A > 2.

definition_p17: Granville and Lumley's S(y), the largest size of an admissible subset of [1, y], which caps the number of primes in an interval of length y, with the bounds y/log y <~ S(y) <~ 2y/log y recorded in Section 4 and the belief that S(y) ~ y/log y.

proposition_1: Granville and Lumley's estimate of the thresholds k_- and k_+ at which the lower and upper tails of a binomial variable with N trials and success probability 1/L fall to 1/x, for N << L log x with L tending to infinity, the probabilistic input of their modified Cramér heuristic.

---

Andrew Granville, Allysa Lumley, "Primes in short intervals: Heuristics and calculations," Experimental Mathematics 32 (2023), no. 2, 378--404, doi:10.1080/10586458.2021.1927256; first circulated as arXiv:2009.05000 (2020).

The edition read for this card is the arXiv v3 manuscript (stamp "arXiv:2009.05000v3 [math.NT] 3 May 2021"); page numbers and labels below are its own. The journal version was not read. The discussion relevant to Problem 1204 is in Section 4 with its subsection 4.1 (pp. 17--19), with the sieve limitation explained in Section 3 (pp. 14--16, the Siegel-zero sentence on p. 15). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2009.05000), every other right reserved.

Admissible sets and the factor-two gap

A finite set of integers is admissible when, for every prime $p$, its elements miss at least one residue class modulo $p$. The paper writes

$$S(y)=\max\{|A|:A\subseteq[1,y]\text{ is admissible}\},$$

equivalently the largest size of an admissible set of length at most $y$; admissibility is invariant under translation. This is the inverse extremal function to Problem 1204's $A(k)$, up to an endpoint shift: $S(y)\geq k$ implies $A(k)\leq y-1$, while $A(k)\leq y$ implies $S(y+1)\geq k$.

Section 4 records

$$\frac{y}{\log y}\lesssim S(y)\lesssim\frac{2y}{\log y}.$$

For the lower bound, translate the primes in $(y,2y]$ into an interval of length $y$; the prime number theorem gives their cardinality. The upper bound is the linear-sieve upper bound (7). On inversion these give the E1204 bounds

$$\left(\frac12-o(1)\right)k\log k \leq A(k)\leq(1+o(1))k\log k.$$

Thus the conjecture $A(k)\sim k\log k$ is equivalent at first order to the paper's belief $S(y)\sim y/\log y$, and the known constants retain a factor-two gap. The paper says a substantial improvement of the sieve upper bound is not currently expected because of the Siegel-zero obstruction: Section 3 cites Granville's "Sieving intervals and Siegel zeros" (its reference [11]) for the result that infinitely many Siegel zeros would make the extremal interval-sieve constants attain the linear-sieve functions $f(u)$ and $F(u)$. This is an obstruction to the method, not a disproof of the coefficient-one conjecture. The paper does not discuss Problem 1204's average-minimization function $B(k)$.

What the prime-interval heuristic does not prove

Let $M(x,y)$ be the maximum number of primes in an interval $(X,X+y]$ with $x<X\leq2x$. Any offsets occupied by those primes form an admissible set, so $M(x,y)\leq S(y)$ when $x\geq y$. Hardy--Littlewood's prime $k$-tuple conjecture would imply, for each fixed $y$, that some translates realize every extremal admissible set, and hence that $\max_{n\geq y}\pi(n,n+y]=S(y)$. Granville and Lumley go further heuristically, predicting $M(x,y)=S(y)$ for $y\leq(1-o(1))\log x$.

Section 4.1 supports that prediction by inserting an extremal admissible set of size $k=S(y)\sim\alpha y/\log y$ into an explicit Hardy--Littlewood prime-tuple heuristic and estimating when a prime translate should first occur. It assumes the unproved asymptotic size of $S(y)$ rather than deriving it. Consequently the short-interval prediction neither constructs admissible sets beyond the translated-prime construction above nor proves $A(k)\sim k\log k$; it also supplies no result about $B(k)$.

Result pages

• Definition and bounds (pp. 3, 17): $S(y)$, the cap $M(x,y)\le S(y)$, and $y/\log y\lesssim S(y)\lesssim2y/\log y$.
• Conjectures (Section 1.7, p. 12): the paper's summary of its conjectures on $M(x,y)$ and $m(x,y)$ in four ranges of $y$.
• Proposition 1 (p. 21): the $1/x$ tail thresholds of a binomial $B(N,1/L)$, the probabilistic input of the modified Cramér heuristic.

Read status. Claims checked: the definitions, bounds, conditional implications, and heuristic qualifications in Sections 3, 4, and 4.1, the summary of conjectures in Section 1.7, and Proposition 1 with its proof were read clause by clause on the page images of the print. The cited sieve and prime-number-theorem arguments were not independently verified.

Bears on. #1204: $S(y)$ inverts $A(k)$ up to an endpoint shift (definition and bounds), so the bounds the paper records for $S(y)$ give $(\frac12-o(1))k\log k\le A(k)\le(1+o(1))k\log k$, and its belief $S(y)\sim y/\log y$ is equivalent at first order to $A(k)\sim k\log k$; the paper proves neither the belief nor anything about $B(k)$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.