Li 2016 lower bound least prime arithmetic progression

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lemma_2_1: Li, Pratt and Shakan's probabilistic heuristic: in a random model of the residue classes of the primes modulo k, with the paper's assumptions (i) to (iii), the thresholds (1 +- epsilon) and (2 +- epsilon) times phi(k) log phi(k) primes decide almost surely how often P(k) is small or large.

theorem_1_1: Li, Pratt and Shakan's theorem that, given epsilon > 0, every large k with at most exp((1/2 - epsilon) log_2 k log_4 k / log_3 k) distinct prime factors has P(k) >> phi(k) log k log_2 k log_4 k / log_3 k, with an effective implied constant; such k have density one.

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Junxian Li, Kyle Pratt, George Shakan, A lower bound for the least prime in an arithmetic progression. arXiv:1607.02543 (2016); published in Q. J. Math. 68 (2017), no. 3, 729--758, doi:10.1093/qmath/hax001. The copy read for this card is arXiv v2 (22 December 2016). The arXiv record names arXiv's non-exclusive distribution license (arXiv:1607.02543), every other right reserved.

For P(k), the maximum over residues l coprime to k of the least prime congruent to l mod k, Theorem 1.1 (printed p. 2) proves: given epsilon > 0, every k > k_0(epsilon) with at most exp((1/2 - epsilon) log_2 k log_4 k / log_3 k) distinct prime factors has P(k) >> phi(k) log k log_2 k log_4 k / log_3 k, with an effective implied constant. These k have density one, so the bound holds for almost every k; it answers a question that Ford, Green, Konyagin, Maynard and Tao had raised. The proof adapts their large-prime-gaps sieve machinery, with the new idea of using sieve weights that capture small multiples of primes as well as primes. The authors also give a heuristic (Lemma 2.1, pp. 4--5) suggesting liminf P(k)/(phi(k) log^2 k) = 1 and limsup P(k)/(phi(k) log^2 k) = 2. For problem 971 the result bounds only the largest least prime P(k) = max_a p(a,k), that is, a single residue class; it says nothing about the positive proportion of classes that problem 971 asks about.

Source: https://arxiv.org/abs/1607.02543.

Read status: claims checked for Theorem 1.1, the density-one argument, the reduction through Lemma 5.1 and the heuristic Lemma 2.1, read clause by clause on the page images of arXiv v2; the proofs were read for structure only. Nothing here is independently reviewed. Result pages: theorem_1_1 and lemma_2_1.

Bears on. #971: Theorem 1.1 (p. 2) gives, for each $k$ it covers, at least one reduced class whose least prime is $\gg\phi(k)\log k\log_2k\log_4k/\log_3k$; it bounds only the maximum $P(k)$ and says nothing about how many classes have a large least prime, which is what the problem asks.

Results.

• Theorem 1.1 (p. 2): given $\epsilon>0$, every $k>k_0(\epsilon)$ with at most $\exp((\frac12-\epsilon)\log_2k\log_4k/\log_3k)$ distinct prime factors has $P(k)\gg\phi(k)\log k\log_2k\log_4k/\log_3k$, with an effective implied constant.
• Lemma 2.1 (pp. 4--5): in a coupon-collector model of the primes' residue classes, the thresholds $(1\pm\epsilon)$ and $(2\pm\epsilon)$ times $\phi(k)\log\phi(k)$ primes decide almost surely how often $P(k)$ is small or large, which suggests $\liminf_kP(k)/(\phi(k)\log^2k)=1$ and $\limsup_kP(k)/(\phi(k)\log^2k)=2$.

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