Pollack 2015 remarks fibers sum divisors function

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corollary_1: Pollack's corollary that the values of the sum-of-divisors function that are the common sigma-value of some amicable tuple, of any length, have density 0 relative to the image of sigma.

theorem_1: Pollack's theorem that for every beta > 0 and every epsilon > 0 there are integers m and n with sigma(m) = sigma(n) and |m/n - beta| < epsilon, answering a 1959 question of Erdős in the affirmative.

theorem_2: Pollack's theorem that the values v of the sum-of-divisors function whose preimages all have the same largest prime factor have density 1 relative to the image of sigma.

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Pollack, Paul, Remarks on fibers of the sum-of-divisors function. In: Analytic Number Theory, Springer, Cham (2015), 305-320, doi:10.1007/978-3-319-22240-0_18. The copy read for this card is the author's manuscript from the author's research page (https://www.pollack-math.net/research.html), which states no terms for the papers it links, and the file prints no notice; the term is unstated.

Pollack records two theorems on the fibers of the sum-of-divisors function sigma. Theorem 1 (p. 1) answers in the affirmative a 1959 question of Erdos (Acta Arith. 5 (1959), p. 172): for every beta > 0 and every epsilon > 0 there are integers m, n with sigma(m) = sigma(n) and |m/n - beta| < epsilon. The proof (Section 2, pp. 2--4) shows that the closure of {log(m/n) : sigma(m) = sigma(n)} is all of R; its main tool is Yitang Zhang's theorem approximating the prime k-tuples conjecture, in the form for general linear forms (Proposition 1, p. 3), through Lemma 1 (p. 3). Remark 2 (p. 4) notes that m and n can be taken coprime. Theorem 2 (p. 2) shows that for asymptotically 100% of the values v in the image of sigma, the density being taken relative to sigma(N), all elements of the fiber sigma^{-1}(v) share the same largest prime factor; the proof (Section 3, pp. 5--14) adapts the methods of Ford and of Ford and Pollack, building on work of Maier and Pomerance. Corollary 1 (p. 2, proved in Section 4, p. 14) deduces that asymptotically 0% of the elements of sigma(N) are the common sigma-value of an amicable tuple, Dickson's generalization of amicable pairs to tuples of any length k.

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Bears on.

• #823: the problem asks whether every alpha >= 1 is the limit of ratios n_k/m_k with sigma(n_k) = sigma(m_k). Theorem 1, applied with beta = alpha and epsilon = 1/k, gives such pairs, so it answers the question yes, for every alpha > 0.

Results.

• Theorem 1 (p. 1): For every beta > 0 and every epsilon > 0 there exist integers m, n with sigma(m) = sigma(n) and |m/n - beta| < epsilon.
• Theorem 2 (p. 2): For asymptotically 100% of the values v in the image of sigma, all elements of sigma^{-1}(v) share the same largest prime factor.
• Corollary 1 (p. 2): Asymptotically 0% of the elements of sigma(N) are the common sigma-value of an amicable tuple, all lengths taken together.

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