Pollack 2016 problems erdos sum divisors function

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conjecture_1_4: The paper conjectures, from a heuristic random model of s(n) = sigma(n) - n, that the numbers outside the range of s have asymptotic density Delta, the limit of (1/log y) times the sum of (1/a)e^{-a/s(a)} over even a up to y; it is not proved.

theorem_1_1: States that the count of up-down reversals and the count of down-up reversals in [1,x] are each at most x/exp((sqrt(3)+o(1))(log_3 x log_4 x)^{1/2}) as x tends to infinity, with log_k the k-fold iterated logarithm.

theorem_1_2: States that the number of up-down reversals in [1,x], the n that are nondeficient with s(n) deficient, is bounded below by a constant times x/((log_2 x)(log_3 x)^3).

theorem_1_3: States that the number of down-up reversals in [1,x], the n that are deficient with s(n) nondeficient, is bounded below by a constant times x/((log_2 x)(log_3 x)^2).

theorem_1_5: States that the number of primitive friendly pairs contained in [1,x], unordered pairs of distinct integers with equal sigma(n)/n and no nontrivial common unitary divisor, is at most x^{1/2+o(1)} as x tends to infinity.

theorem_1_6: States that for each fixed nonnegative integer k the number N_k(x) of n at most x having at least k friends m at most x is (alpha_k+o(1))x for a constant alpha_k, and that alpha_k tends to 0 as k tends to infinity.

theorem_5_2: States that the limiting proportions alpha_k of Theorem 1.6 decrease strictly while positive: if alpha_k > 0, then alpha_k > alpha_{k+1}.

theorem_p24: States that the number g(x) of coprime pairs a < b <= x with sigma(a) = sigma(b) exceeds x^{1.4} for all sufficiently large x, so g(x)/x tends to infinity.

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Paul Pollack, Carl Pomerance, Some problems of Erdős on the sum-of-divisors function. Transactions of the American Mathematical Society, Series B 3 (2016), 1-26. doi:10.1090/btran/10. The file prints "©2016 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)": the Creative Commons Attribution-NonCommercial 3.0 license; its offprint footer also reads "This is a free offprint provided to the author by the publisher. Copyright restrictions may apply."

The paper takes up three of Erdős's themes on $\sigma(n)$ and $s(n)=\sigma(n)-n$, and closes with a fourth. First, aliquot reversals: an up-down reversal is a nondeficient $n$ with $s(n)$ deficient, a down-up reversal a deficient $n$ with $s(n)$ nondeficient (p. 2). Theorem 1.1 lowers the upper bound for both counts in $[1,x]$ to $x/\exp((\sqrt3+o(1))(\log_3x\,\log_4x)^{1/2})$, while Theorems 1.2 and 1.3 give the first lower bounds, $\gg x/((\log_2x)(\log_3x)^3)$ and $\gg x/((\log_2x)(\log_3x)^2)$, established for even numbers (p. 3). Second, Conjecture 1.4 predicts, from a heuristic random model of $s$, the asymptotic density $\Delta$ of the nonaliquot numbers (those outside the range of $s$); the counts to $10^{10}$ in Table 1 are consistent with a density of about $0.17$ (p. 3), and Section 3.3 (p. 17) treats the analogue for $n-\varphi(n)$. Third, on friendly numbers (equal $\sigma(n)/n$), Theorem 1.5 bounds the primitive friendly pairs in $[1,x]$ by $x^{1/2+o(1)}$, strengthening the convergence of $\sum1/m$ used by Erdős, and Theorem 1.6 shows that the number $N_k(x)$ of $n\le x$ with at least $k$ friends in $[1,x]$ is $(\alpha_k+o(1))x$ with $\alpha_k\to0$; Theorem 5.2 shows $\alpha_k>\alpha_{k+1}$ whenever $\alpha_k>0$. Finally, Section 6 (pp. 23--24) proves that the number of coprime pairs $a<b\le x$ with $\sigma(a)=\sigma(b)$ exceeds $x^{1.4}$ for large $x$. The reversal upper bounds rest on sieve methods, counts of primitive nondeficient numbers and bounds for solutions of $\sigma(n)\equiv a\pmod n$; the lower bounds on explicit sieve constructions.

No result of the paper bears directly on Problem 410, which iterates $\sigma$ rather than $s$; the paper is context for it. It records that $n$ and $s(n)$ differ in parity exactly when $n$ is a square or twice a square (Section 1, p. 2), and its reversal counts measure how often iteration of $s$ turns between nondeficient and deficient.

Source: https://math.dartmouth.edu/~carlp/btran10.pdf.

Results

Page numbers are those of the journal print (pp. 1--26).

• Theorem 1.1 (p. 2): both reversal counts in $[1,x]$ are at most $x/\exp((\sqrt3+o(1))(\log_3x\,\log_4x)^{1/2})$.
• Theorem 1.2 (p. 2): up-down reversals in $[1,x]$ number $\gg x/((\log_2x)(\log_3x)^3)$.
• Theorem 1.3 (p. 2): down-up reversals in $[1,x]$ number $\gg x/((\log_2x)(\log_3x)^2)$.
• Conjecture 1.4 (p. 3): the nonaliquot numbers have density $\Delta=\lim_{y\to\infty}(\log y)^{-1}\sum_{a\le y,\,2\mid a}a^{-1}e^{-a/s(a)}$; a heuristic, with the $\varphi$ analogue of Section 3.3 (p. 17).
• Theorem 1.5 (p. 4): at most $x^{1/2+o(1)}$ primitive friendly pairs lie in $[1,x]$.
• Theorem 1.6 (p. 5): for each fixed $k\ge0$, $N_k(x)=(\alpha_k+o(1))x$, and $\alpha_k\to0$.
• Theorem 5.2 (p. 22): if $\alpha_k>0$ then $\alpha_k>\alpha_{k+1}$.
• Section 6 result (pp. 23--24): coprime pairs $a<b\le x$ with $\sigma(a)=\sigma(b)$ number more than $x^{1.4}$ for large $x$.

Read status. Claims checked for the eight results above, read clause by clause on the print; the proofs were read for their structure only, and the computations behind Tables 1--4 were not rerun.

Bears on

• Problem 824: the Section 6 result gives $h(x)>(x-1)^{1.4}$ for large $x$, hence $h(x)/x\to\infty$; the problem asks whether $h(x)>x^{2-o(1)}$, which the paper does not decide.
• Problem 418: context only. Section 3.3 (p. 17) gives a conjectural density and computed counts for the integers not of the form $n-\varphi(n)$; they prove nothing about the problem.
• Problem 830: context only. The lesser member of an amicable pair is an up-down reversal (p. 2), so Theorem 1.1 bounds above the number of lesser members of amicable pairs in $[1,x]$; the problem asks for infinitely many pairs and a lower bound, which an upper bound does not decide.
• Problem 410: context only, as explained above.

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