Erdos 1980 asymptotic formulas generalized divisor functions

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corollary_1: Erdős and Sárközy's corollary that for every Ω > 0 and every large x, a sequence whose reciprocal sum up to x exceeds a constant c_4(Ω) has some n up to y = x^{1+1/(f_A(x))^{1/4}} with more than Ω f_A(x) of its members as divisors, with no condition on where the members lie.

corollary_2: Erdős and Sárközy's corollary that for every Ω > 1 and every large x, a sequence whose reciprocal sum up to x exceeds a constant c_5(Ω) but whose divisor counts up to x stay at most Ω f_A(x) has more than x^{1-1/(f_A(x))^{1/3}} members up to x.

problem_2: The paper's Problem 2 asks whether for every Ω > 0 there are constants c_25(Ω) and X_7(Ω) such that x > X_7 and f_A(x) > c_25 imply D_A(x^2) > (f_A(x))^Ω; the paper leaves it open.

theorem_1: The theorem of Part III of the Erdős--Sárközy series, restated as Theorem 1 of Part IV, that for every Ω > 0 and every large x a sequence whose reciprocal sum up to x exceeds (log log x)^20 has some n up to x with more than Ω times that sum of its members as divisors.

theorem_2: Erdős and Sárközy's theorem that for every Ω > 1 and every large x, a sequence whose reciprocal sum up to x exceeds a constant c_3(Ω) and which has no member in the interval from x^{1-1/(f_A(x))^{1/3}} to x has some n up to x with more than Ω times that sum of its members as divisors.

theorem_3: Erdős and Sárközy's construction showing that Theorem 2 fails when its exponent 1 - 1/(f_A(x))^{1/3} is replaced by 1 - 1/c^{f_A(x)}: for large x and c_6 < t < c_7 log log x there is a sequence with reciprocal sum of order t, no member in (x^{1-1/c_10^{f_A(x)}}, x] and D_A(x) < c_11 f_A(x).

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P. Erdős, A. Sárközy: Some asymptotic formulas on generalized divisor functions, IV., Studia Sci. Math. Hungar. 15 (1980) no. 4, 467--479 (MR 84m:10038c; Zbl 512.10037); the print records "(Received September 25, 1981)" (p. 479).

This fourth part of the Erdos-Sarkozy series studies D_A(x) = max_{n <= x} d_A(n), where d_A(n) counts the elements of a sequence A that divide n, and f_A(x) is the sum of 1/a over a in A with a <= x (the count of those a is N_A(x)). The introduction reviews Parts I to III: Part I gave lim sup D_A(x)/f_A(x) = infinity for every infinite A, Part II sharpened this to lim sup D_A(x)/exp(c_1 (log f_A(x))^2) = infinity whenever f_A(x) -> infinity, and Part III proved (Theorem 1 here) that for every Omega > 0 and x > X_0(Omega), f_A(x)

(log log x)^{20} implies D_A(x) > Omega f_A(x). The aim of this paper is to find a function y = y(x) as small as possible such that f_A(x) -> infinity forces D_A(y(x))/f_A(x) -> infinity; Theorem 2 states that for every Omega > 1 there are constants c_3(Omega), X_1(Omega) such that for x > X_1, f_A(x) > c_3 together with [x^{1-1/(f_A(x))^{1/3}}, x] having no element of A implies D_A(x) Omega f_A(x). Corollary 1 drops the interval condition at the price of moving the conclusion to y = x^{1+1/(f_A(x))^{1/4}}: for every Omega > 0, x > X_2(Omega) and f_A(x) > c_4(Omega) give D_A(y) > Omega f_A(x). Theorem 3 shows that the exponent 1 - 1/(f_A(x))^{1/3} in Theorem 2 cannot be replaced by 1 - 1/c_6^{f_A(x)}. The proof of Theorem 2 splits A according to whether a member has a divisor in a range of moderate size and builds integers up to x with many divisors in A; its last case uses two lemmas from Part III on integers with unusually few or unusually many prime factors in a range, consequences of a result of K. K. Norton. Section 5 (pp. 478--479) poses two problems on D_A(x^2). The paper bears on the cited problem, whether max_{n<x} d_A(n) exceeds every fixed power of f_A(x) infinitely often, through the Part II theorem (1) its introduction restates (p. 468), which answers it yes for every power when f_A(x) -> infinity, with the restated Part I result covering a bounded f_A(x); both are proved in the earlier parts, and this paper's own theorems give only multiples Omega f_A(x).

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Bears on. #444: the paper's introduction restates (p. 468) the Part II theorem (1), that $f_A(x)\to+\infty$ implies $\limsup D_A(x)/\exp(c_1(\log f_A(x))^2)=+\infty$, and the Part I theorem $\limsup D_A(x)/f_A(x)=+\infty$ for every infinite $A$, the results the problem's claim page records as answering the question yes for every $k$ (the paper takes $n\le x$ and $a\le x$ where the problem takes $n<x$ and $a<x$). Both are proved in the earlier parts, not here. This paper's own results, Theorem 2 and Corollary 1, give only multiples $\Omega f_A(x)$, and its Problem 2 (p. 479), whether $f_A(x)>c_{25}(\Omega)$ forces $D_A(x^2)>(f_A(x))^{\Omega}$ for all large $x$, is posed and left open.

Results.

• Theorem 1 (p. 468, from Part III): for every $\Omega>0$ and $x>X_0(\Omega)$, $f_A(x)>(\log\log x)^{20}$ implies $D_A(x)>\Omega f_A(x)$.
• Theorem 2 (p. 468): for every $\Omega>1$ there are $c_3(\Omega)$, $X_1(\Omega)$ such that $x>X_1$, $f_A(x)>c_3$ and $\bigl[x^{1-1/(f_A(x))^{1/3}},x\bigr]\cap A=\emptyset$ imply $D_A(x)>\Omega f_A(x)$.
• Corollary 1 (pp. 468--469): for every $\Omega>0$ there are $c_4(\Omega)$, $X_2(\Omega)$ such that $x>X_2$ and $f_A(x)>c_4$ imply $D_A(y)>\Omega f_A(x)$ with $y=x^{1+1/(f_A(x))^{1/4}}$.
• Corollary 2 (p. 469): for every $\Omega>1$ there are $c_5(\Omega)$, $X_3(\Omega)$ such that $x>X_3$, $f_A(x)>c_5$ and $D_A(x)\le\Omega f_A(x)$ imply $N_A(x)>x^{1-1/(f_A(x))^{1/3}}$.
• Theorem 3 (p. 469): for absolute constants, $x>X_4$ and $c_6<t<c_7\log\log x$, some $A$ has $c_8t<f_A(x)<c_9t$, no member in $\bigl(x^{1-1/c_{10}^{f_A(x)}},x\bigr]$ and $D_A(x)<c_{11}f_A(x)$.
• Problem 2 (p. 479): whether $x>X_7(\Omega)$ and $f_A(x)>c_{25}(\Omega)$ imply $D_A(x^2)>(f_A(x))^{\Omega}$ for every $\Omega>0$; open in the paper.

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