Bergelson 2017 density coprime tuples form where are

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theorem_1: The paper's theorem that for f in a Hardy field with log t log_4 t below f and f strictly between two consecutive powers of t, the integers n coprime to the integer part of f(n) have natural density 6/pi^2.

theorem_2: The paper's theorem that for f_1, ..., f_k in a Hardy field meeting its growth conditions and with each ratio f_{i+1}/f_i above (log log t)^4, the integers n with gcd of n and the integer parts of the f_i(n) equal to one have natural density 1/zeta(k+1).

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Bergelson, Vitaly and Richter, Florian Karl, On the density of coprime tuples of the form {$(n,\lfloor f_1(n)\rfloor,\dots,\lfloor f_k(n)\rfloor )$}, where {$f_1,\dots,f_k$} are functions from a {H}ardy field. In: Number Theory – Diophantine Problems, Uniform Distribution and Applications, Springer, Cham (2017), 109--135. DOI: 10.1007/978-3-319-55357-3_5.

The paper extends the classical fact that gcd(n,m) = 1 with probability 6/pi^2 to tuples built from smooth slowly varying functions. Writing f ≺ g when g(t)/f(t) tends to infinity, Theorem 1 (p. 3) shows that if f lies in a Hardy field and satisfies the growth conditions (A) log(t) log_4(t) ≺ f(t), where log_4 is the fourth iterated logarithm, and (B) t^{j-1} ≺ f(t) ≺ t^j for some j in N, then the natural density of the set of n with gcd(n, floor(f(n))) = 1 exists and equals 6/pi^2; the examples listed are n^c for non-integral c, log^2 n, n^{sqrt 3} log n, n/log_2 n, log(n!), Li(n) and log|B_{2n}|. Theorem 2 (p. 4) is the k-dimensional version: if f_1,...,f_k lie in a Hardy field and satisfy (A), (B) and the separation condition (C) f_{i+1}/f_i ≻ log_2^4(t) for i = 1,...,k-1, that is, each ratio grows faster than (log log t)^4, then the density of the set of n with gcd(n, floor(f_1(n)),...,floor(f_k(n))) = 1 exists and equals 1/zeta(k+1). The paper proves Theorem 2, of which Theorem 1 is the case k = 1. The proof establishes differential inequalities for Hardy-field functions, applies van der Corput's method to the resulting exponential sums, converts these into discrepancy estimates, and runs a Möbius inclusion-exclusion over divisors (Proposition 16, p. 15, applied in Corollary 17, p. 18); when f_1 grows more slowly than t/log_2 t it argues instead through an equidistribution theorem of Boshernitzan and an estimate after Erdős and Lorentz (Theorem 21, p. 22). The introduction traces the problem to Watson's case f(n) = n alpha with alpha irrational, attributes the case f(n) = n^c with c > 0 non-integral to Lambek and Moser (Canad. J. Math. 7 (1955) 155--158, for 0 < c < 1) and Delmer and Deshouillers (Period. Math. Hungar. 45 (2002) 15--20, the general case), and cites Erdős and Lorentz, quoting their heuristic that coprimality should persist whenever g(n) does not preserve arithmetic properties of n; Section 7 (pp. 23--24) lists four open questions, among them whether (C) can be weakened to f_{i+1}/f_i ≻ 1 and whether (B) can be weakened. For Erdős problem 1149, the paper lists n^c with c not an integer among the functions to which Theorem 1 applies (p. 3), and f(t) = t^alpha with alpha > 0 not an integer meets (A) and (B) with j = ceil(alpha); the introduction credits this case to Lambek and Moser and to Delmer and Deshouillers.

Source: https://arxiv.org/abs/1611.08044. The copy read for this card is the arXiv preprint arXiv:1611.08044v2 (20 May 2017), whose title page carries the date October 3, 2018; the pages cited on this card and its result pages are the preprint's, numbered 1 to 26. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1611.08044), every other right reserved.

Bears on. #1149: Theorem 1 (p. 3) with f(t) = t^alpha gives density 6/pi^2 for the integers n with gcd(n, floor(n^alpha)) = 1, for every alpha > 0 that is not an integer.

Results.

• Theorem 1 (p. 3): for f in a Hardy field satisfying (A) and (B), the set {n : gcd(n, floor(f(n))) = 1} has natural density 6/pi^2.
• Theorem 2 (p. 4): for f_1,...,f_k in a Hardy field satisfying (A), (B) and (C), the set {n : gcd(n, floor(f_1(n)),...,floor(f_k(n))) = 1} has natural density 1/zeta(k+1).

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