Finding narrow admissible tuples

For any natural number [math]\displaystyle{ k_0 }[/math], an admissible [math]\displaystyle{ k_0 }[/math]-tuple is a finite set [math]\displaystyle{ {\mathcal H} }[/math] of integers of cardinality [math]\displaystyle{ k_0 }[/math] which avoids at least one residue class modulo [math]\displaystyle{ p }[/math] for each prime [math]\displaystyle{ p }[/math]. (Note that one only needs to check those primes [math]\displaystyle{ p }[/math] of size at most [math]\displaystyle{ k_0 }[/math], so this is a finitely checkable condition.) Let [math]\displaystyle{ H(k_0) }[/math] denote the minimal diameter [math]\displaystyle{ \max {\mathcal H} - \min {\mathcal H} }[/math] of an admissible [math]\displaystyle{ k_0 }[/math]-tuple. As part of the Polymath8 project, we would like to find as good an upper bound on [math]\displaystyle{ H(k_0) }[/math] as possible for given values of [math]\displaystyle{ k_0 }[/math]. To a lesser extent, we would also be interested in lower bounds on this quantity.

Upper bounds

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo [math]\displaystyle{ p }[/math] from an interval for a lot of primes [math]\displaystyle{ p }[/math]. Examples of sieves, in roughly increasing order of efficiency, are listed below.

Zhang sieve

The Zhang sieve uses the tuple

[math]\displaystyle{ {\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\} }[/math]

where [math]\displaystyle{ m }[/math] is taken to optimize the diameter [math]\displaystyle{ p_{m+k_0}-p_{m+1} }[/math] while staying admissible (in practice, this basically means making [math]\displaystyle{ m }[/math] as small as possible). Certainly any [math]\displaystyle{ m }[/math] with [math]\displaystyle{ p_{m+1} \gt k_0 }[/math] works, but this is not optimal. Applying the prime number theorem then gives the upper bound [math]\displaystyle{ H \leq (1+o(1)) k_0\log k_0 }[/math].

Hensley-Richards sieve

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

[math]\displaystyle{ {\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\} }[/math]

where m is again optimised to minimize the diameter while staying admissible.

Asymmetric Hensley-Richards sieve

The asymmetric Hensley-Richard sieve uses the tuple

[math]\displaystyle{ {\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\} }[/math]

where [math]\displaystyle{ i }[/math] is an integer and [math]\displaystyle{ i,m }[/math] are optimised to minimize the diameter while staying admissible.

Schinzel sieve

Given [math]\displaystyle{ 0\lt y\lt z }[/math], the Schinzel sieve (discussed in [HR1973], [CJ2001] first sieves by [math]\displaystyle{ 1\bmod p }[/math] for primes [math]\displaystyle{ p \le y }[/math] and by [math]\displaystyle{ 0\bmod p }[/math] for primes [math]\displaystyle{ y \lt p \le z }[/math]. For a given choice of [math]\displaystyle{ y }[/math], the parameter [math]\displaystyle{ z }[/math] is minimized subject to ensuring that the first [math]\displaystyle{ k_0 }[/math] survivors (after the first) form an admissible sequence [math]\displaystyle{ \mathcal{H} }[/math], so the only free parameter is [math]\displaystyle{ y }[/math], which is chosen to minimize the diameter of [math]\displaystyle{ \mathcal{H} }[/math]. The case [math]\displaystyle{ y=1 }[/math] corresponds to a sieve of Eratosthenes, which will typically yield the same sequence as Zhang with the minimal (but not necessarily optimal) value of [math]\displaystyle{ m }[/math] that yields an admissible [math]\displaystyle{ k_0 }[/math]-tuple. As originally proposed, the Schinzel sieve works over the positive integers, but one can instead sieve intervals centered about the origin, or asymmetric intervals, as with the Hensley-Richards sieve.

Greedy sieve

For a given interval (e.g., [math]\displaystyle{ [1,x] }[/math], [math]\displaystyle{ [-x,x] }[/math], or asymmetric [math]\displaystyle{ [x_0,x_1] }[/math]) one sieves a single residue class [math]\displaystyle{ a \bmod p }[/math] for increasing primes [math]\displaystyle{ p=2,3,5,\ldots }[/math], with [math]\displaystyle{ a }[/math] chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize [math]\displaystyle{ a\in[0,p-1] }[/math], maximize [math]\displaystyle{ a\in [0,p-1] }[/math], minimize [math]\displaystyle{ |a-\lfloor p/2\rfloor| }[/math], or randomly. If not all residue classes modulo [math]\displaystyle{ p }[/math] are occupied by survivors, then [math]\displaystyle{ a }[/math] will be chosen so that no survivors are sieved. This necessarily occurs once [math]\displaystyle{ p }[/math] exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest [math]\displaystyle{ k_0 }[/math]-tuple [math]\displaystyle{ {\mathcal H} }[/math] among the survivors (if there are fewer than [math]\displaystyle{ k_0 }[/math] survivors, retry with a wider interval).

Greedy-Schinzel sieve

Heuristically, the performance of the greedy sieve is significantly improved by starting with a Schinzel sieve with [math]\displaystyle{ y=2 }[/math] and [math]\displaystyle{ z=\sqrt{x_1-x_0} }[/math] and then continuing in a greedy fashion (this method was proposed by Sutherland and originally referred to as a "greedy-greedy" approach).

Seeded greedy sieve

Given an initial sequence [math]\displaystyle{ {\mathcal S} }[/math] that is known to contain an admissible [math]\displaystyle{ k_0 }[/math]-tuple, one can apply greedy sieving to the minimal interval containing [math]\displaystyle{ {\mathcal S} }[/math] until an admissible sequence of survivors remains, and then choose the narrowest [math]\displaystyle{ k_0 }[/math]=tuple it contains. The sieving methods above can be viewed as the special case where [math]\displaystyle{ {\mathcal S} }[/math] is the set of integers in some interval. The main difference is that the choice of [math]\displaystyle{ {\mathcal S} }[/math] affects when ties occur and how they are broken with greedy sieving. One approach is to take [math]\displaystyle{ {\mathcal S} }[/math] to be the union of two [math]\displaystyle{ k_0 }[/math]-tuples that lie in roughly the same interval (see Iterated merging) below.

Iterated merging

Given an admissible [math]\displaystyle{ k_0 }[/math]-tuple [math]\displaystyle{ \mathcal{H}_1 }[/math], one can attempt to improve it using an iterated merging approach suggested by Castryck. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible [math]\displaystyle{ k_0 }[/math]-tuple [math]\displaystyle{ \mathcal{H}_2 }[/math] in roughly the same interval as [math]\displaystyle{ \mathcal{H}_1 }[/math], then performs a randomized greedy sieve using the seed set [math]\displaystyle{ \mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2 }[/math] to obtain an admissible [math]\displaystyle{ k_0 }[/math]-tuple [math]\displaystyle{ \mathcal{H}_3 }[/math]. If [math]\displaystyle{ \mathcal{H}_3 }[/math] is narrower than [math]\displaystyle{ \mathcal{H}_2 }[/math], replace [math]\displaystyle{ \mathcal{H}_2 }[/math] with [math]\displaystyle{ \mathcal{H}_3 }[/math], otherwise try again with a new [math]\displaystyle{ \mathcal{H}_3 }[/math]. Eventually the diameter of [math]\displaystyle{ \mathcal{H}_2 }[/math] will become less than or equal to that of [math]\displaystyle{ \mathcal{H}_1 }[/math]. As long as [math]\displaystyle{ \mathcal{H}_1\ne \mathcal{H}_2 }[/math], one can continue to attempt to improve [math]\displaystyle{ \mathcal{H}_2 }[/math], but in practice one stops after some number of retries.

As described by Sutherland, one can then replace [math]\displaystyle{ \mathcal{H}_1 }[/math] with [math]\displaystyle{ \mathcal{H}_2 }[/math] and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing [math]\displaystyle{ \mathcal{H}_2 }[/math], which is typically made wider than the minimal interval containing [math]\displaystyle{ \mathcal{H}_1 }[/math] by a small factor [math]\displaystyle{ \delta }[/math] on each side (Sutherland suggests [math]\displaystyle{ \delta = 0.0025 }[/math]), and the number of failed attempts allowed while attempting to impove [math]\displaystyle{ \mathcal{H}_2 }[/math].

Eventually this process will tend to converge to particular [math]\displaystyle{ \mathcal{H}_1 }[/math] that it cannot improve (or more generally, a set of similar [math]\displaystyle{ \mathcal{H}_1 }[/math]'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

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Iterated merging can be viewed as a form of simulated annealing. The set [math]\displaystyle{ \mathcal{S} }[/math] initially contains at least two admissible [math]\displaystyle{ k_0 }[/math]-tuples (typically many more), and as the algorithm proceeds the set [math]\displaystyle{ \mathcal{S} }[/math] converges toward [math]\displaystyle{ \mathcal{H}_1 }[/math] and the number of admissible [math]\displaystyle{ k_0 }[/math]-tuples it contains declines. One can regard the cardinality of the difference between [math]\displaystyle{ \mathcal{S} }[/math] and [math]\displaystyle{ \mathcal{H}_1 }[/math] as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given [math]\displaystyle{ \mathcal{S} }[/math]).

Lower bounds

There is a substantial amount of literature on bounding the quantity [math]\displaystyle{ \pi(x+y)-\pi(x) }[/math], the number of primes in a shifted interval [math]\displaystyle{ [x+1,x+y] }[/math], where [math]\displaystyle{ x,y }[/math] are natural numbers. As a general rule, whenever a bound of the form

[math]\displaystyle{ \pi(x+y) - \pi(x) \leq F(y) }[/math] (*)

is established for some function [math]\displaystyle{ F(y) }[/math] of [math]\displaystyle{ y }[/math], the method of proof also gives a bound of the form

[math]\displaystyle{ k_0 \leq F( H(k_0)+1 ). }[/math] (**)

Indeed, if one assumes the prime tuples conjecture, any admissible [math]\displaystyle{ k_0 }[/math]-tuple of diameter [math]\displaystyle{ H }[/math] can be translated into an interval of the form [math]\displaystyle{ [x+1,x+H+1] }[/math] for some [math]\displaystyle{ x }[/math]. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for [math]\displaystyle{ x\gt y }[/math], the set of primes between [math]\displaystyle{ x+1 }[/math] and [math]\displaystyle{ x+y }[/math] is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

Brun-Titchmarsh inequality

The Brun-Titchmarsh theorem gives

[math]\displaystyle{ \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y} }[/math]

which then gives the lower bound

[math]\displaystyle{ H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0 }[/math].

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

[math]\displaystyle{ k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}. }[/math]

First Montgomery-Vaughan large sieve inequality

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

[math]\displaystyle{ k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2 }[/math]

for any [math]\displaystyle{ Q \gt 1 }[/math], which is a parameter that one can optimise over (the optimal value is comparable to [math]\displaystyle{ H(k_0)^{1/2} }[/math]).

Second Montgomery-Vaughan large sieve inequality

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

[math]\displaystyle{ k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1} }[/math]

for any [math]\displaystyle{ z \gt 1 }[/math], which is a parameter similar to [math]\displaystyle{ Q }[/math] in the previous inequality, and [math]\displaystyle{ c }[/math] is an absolute constant. In the original paper of Montgomery and Vaughan, [math]\displaystyle{ c }[/math] was taken to be [math]\displaystyle{ 3/2 }[/math]; this was then reduced to [math]\displaystyle{ \sqrt{22}/\pi }[/math] [B1995, p.162] and then to [math]\displaystyle{ 3.2/\pi }[/math] [M1978]. It is conjectured that [math]\displaystyle{ c }[/math] can be taken to in fact be [math]\displaystyle{ 1 }[/math].

Benchmarks

[math]\displaystyle{ k_0 }[/math]	3,500,000	34,429	26,024
Upper bounds
Zhang sieve	59,093,364	411,932	303,558
Hensley-Richards sieve	57,554,086	402,790	297,454
Asymmetric Hensley-Richards		401,700	297,076
Schinzel sieve
Best known tuple	57,554,086	386,750	285,210
Lower bounds
MV with [math]\displaystyle{ c=1 }[/math]		234,642	172,924
MV with [math]\displaystyle{ c=3.2/\pi }[/math]		234,322	172,719
MV with [math]\displaystyle{ c=\sqrt{22}/\pi }[/math]		227,078	167,860
Second Montgomery-Vaughan		226,987	167,793
Brun-Titchmarsh		211,046	155,555
First Montgomery-Vaughan		196,729	145,711