Almost primes and the Banks–Martin conjecture

Lichtman, Jared Duker

doi:10.1016/j.jnt.2019.11.006

Cited by 13 publications

(22 citation statements)

References 8 publications

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“…where I k,q represents the set of monic polynomials in F q [x] with k irreducible factors counted with multiplicity. Analogously to the observation of Lichtman [8], we find that the conjecture is false for q = 2, 3, and 4 by direct numerical computation in Section 6.3. However, in this section we will show that for each k, there exists q k such that the inequality holds up to F (I k,q ) for all q ≥ q k , and we will establish an upper bound on the size of q k .…”

Section: The Banks-martin Inequalitysupporting

confidence: 81%

On the size of primitive sets in function fields

Gómez-Colunga

Kavaler

McNew

et al. 2020

Finite Fields and Their Applications

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A set is primitive if no element of the set divides another. We consider primitive sets of monic polynomials over a finite field and find natural generalizations of many of the results known for primitive sets of integers. In particular we generalize a result of Besicovitch to show that there exist primitive sets in Fq[x] with upper density arbitrarily close to q−1 q . Then, for a primitive set A, we consider the sum a∈A 1 q deg a deg a , the natural analogue in this setting of a sum considered by Erdős for primitive subsets of the integers, and show that it is uniformly bounded over all primitive sets A. We end with a generalization of work of Martin and Pomerance on the asymptotic growth rate of the counting function of a primitive set. Along the way we prove a quantitative analogue of the Hardy-Ramanujan theorem for function fields, as well as bounds on the size of the k-th irreducible polynomial.

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Section: The Banks-martin Inequalitysupporting

confidence: 81%

On the size of primitive sets in function fields

Gómez-Colunga

Kavaler

McNew

et al. 2020

Finite Fields and Their Applications

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“…Here the interchange of sum and integral holds by Tonelli's theorem, since f (N k , h) ≤ f (N k ) converges uniformly after Erdős. The significance of the identity (2.1) was first observed when k = 1, h = 0 by H. Cohen [2, p.6], who rapidly computed f (N 1 ) = 1.636616 In general for k ≥ 1, Proposition 3.1 in [6] gives an explicit formula for P k in terms of P ,…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…We have already verified the claim directly for k ≤ 20, since in this case h k ≤ h 2 = 1.04 • • • . For k > 20, the proof strategy is similar to that of Theorem 5.5 in [6]. That is, the integral f (N k , h) = ∞ 1 P k (s)e (1−s)h ds has its mass concentrated near 1 as k → ∞, so it suffices to truncate the integration to [1, 1.01] say, as a lower bound.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Translated sums of primitive sets

Lichtman

2021

Preprint

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The Erdős primitive set conjecture states that the sum f (A) = a∈A 1 a log a , ranging over any primitive set A of positive integers, is maximized by the set of prime numbers. Recently Laib, Derbal, and Mechik proved that the translated Erdős conjecture for the sum f (A, h) = a∈A 1 a(log a+h) is false starting at h = 81, by comparison with semiprimes. In this note we prove that such falsehood occurs already at h = 1.04 • • • , and show this translate is best possible for semiprimes. We also obtain results for translated sums of k-almost primes with larger k.

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“…uniformly for x ≥ 3 and k ≤ (2 − δ) log 2 x. Then by partial summation, ( 6) implies (5) with constant coefficient…”

Section: Introductionmentioning

confidence: 99%

Higher Mertens constants for almost primes

Bayless,

Kinlaw,

Lichtman

2021

Preprint

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For k ≥ 1, a k-almost prime is a positive integer with exactly k prime factors, counted with multiplicity. In this article we give elementary proofs of precise asymptotics for the reciprocal sum of k-almost primes. Our results match the strength of those of classical analytic methods. We also study the limiting behavior of the constants appearing in these estimates, which may be viewed as higher analogues of the Mertens constant β = 0.2614... Further, in the case k = 2 of semiprimes we give yet finer-scale and explicit estimates, as well as a conjecture.

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Almost primes and the Banks–Martin conjecture

Cited by 13 publications

References 8 publications

On the size of primitive sets in function fields

On the size of primitive sets in function fields

Translated sums of primitive sets

Higher Mertens constants for almost primes

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