Erdos 1952 sum

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theorem: Erdős's theorem that for an irreducible integer polynomial f, positive on the positive integers, there are constants c_1, c_2 > 0 depending on f with c_1 x log x < sum_{k <= x} d(f(k)) < c_2 x log x for all x >= 2.

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Erdős, P., On the sum {$\sum^x_{k=1} d(f(k))$}. J. London Math. Soc. 27 (1952), 7--15. No copyright or license line is printed in the file (pp. 7--8 and 14--15 read); the Wiley article page could not be read, and the article's Crossref record (DOI 10.1112/jlms/s1-27.1.7, read 2026-10-02) names Wiley as the journal's publisher and lists only Wiley's text-and-data-mining terms and its version-of-record terms and conditions (onlinelibrary.wiley.com/termsAndConditions#vor), no open license; every other right reserved.

For $d(n)$ the divisor function and $f$ an irreducible polynomial of degree $l$ with integer coefficients, assumed positive on the positive integers, the paper's single Theorem (p. 7) proves that there are positive constants $c_1,c_2$, depending on $f$, with $c_1x\log x<\sum_{k\le x}d(f(k))<c_2x\log x$ for $x\ge2$. The paper calls the lower bound not difficult and known, citing Bellman, Duke Math. J. 17 (1950), 159--168, and proves it in Sections 4 and 5 (pp. 14--15) by showing that the sum of $d_x(f(k))$, the number of divisors of $f(k)$ not exceeding $x$, is already greater than $c_1x\log x$; it says the asymptotic $c_3x\log x+o(x\log x)$ for that restricted sum, its (2), would not be hard to show, without proving it. The upper bound is the harder part, since $d(f(k))$ is not easily bounded in terms of $d_x(f(k))$. The paper says this can be done for $l=2$, where Bellman and Shapiro proved, unpublished, the asymptotic $c_4x\log x+o(x\log x)$, its (3); it says (3) very likely holds for $l>2$ as well, but that it cannot prove this (p. 7).

The tools are Lemmas 1 to 9 of Section 2 (pp. 8--10) and Lemma 10 of Section 4 (p. 14). Lemma 1 is van der Corput's second-moment bound $\sum_{k\le x}d(f(k))^2<x(\log x)^{c_5}$, and Lemma 2 derives from it, by Schwarz's inequality, that $\sum_{i\le t}d(f(k_i))<x$ for any $t$ distinct positive integers $k_i<x$ with $t<x(\log x)^{-c_5}$. With $\rho(a)$ the number of solutions of $f(k)\equiv0\pmod a$, $0\le k<a$, and $D$ the discriminant of $f$, Lemma 3 records that $\rho$ is multiplicative, that $\rho(p^\alpha)\le l$ when $p\nmid D$, Nagell's $\rho(p^\alpha)=\rho(p^{2\sigma+1})$ when $p^\sigma\,\|\,D$ and $\alpha>2\sigma$, and that $\rho(p^\alpha)\le c_6$ always (Nagell: $c_6=lD^2$ suffices). Lemma 4 says that for $1\le u\le x$ the number $N$ of $k$ with $1\le k\le x$ and $f(k)\equiv0\pmod u$ satisfies $\frac{x}{2u}\rho(u)\le N\le\frac{2x}{u}\rho(u)$. Lemma 10 gives $\sum_{k\le y}\rho(k)>c_{13}y$ for large $y$. The paper also states, without proof, that its upper-bound method combined with Brun's method would give $\sum_{p\le x}d(f(p))=O(x)$ over primes, answering a question in Bellman's paper (p. 7).

Problem 975 asks whether $\sum_{n\le X}\tau(f(n))\sim cX\log X$. The paper gives the order of magnitude for every $f$ in its setting and the asymptotic for none; it reports the degree-two case only as Bellman and Shapiro's unpublished result.

Source: https://users.renyi.hu/~p_erdos/Erdos.html.

Read status: claims checked for the Theorem, (2), (3), the remark on primes and Lemmas 1 to 4 and 10, read clause by clause on the page images of the print; the proofs read for structure only. Nothing here is independently reviewed. Result page: theorem.

Bears on. #975: the Theorem (p. 7) proves $c_1x\log x<\sum_{k\le x}d(f(k))<c_2x\log x$ for irreducible $f$ positive on the positive integers, the order of magnitude of the sum whose asymptotic the problem asks for; it proves no asymptotic for any $f$.

Results.

• Theorem (p. 7): for irreducible $f$ of degree $l$ with $f(k)>0$ for $k\ge1$, there are $c_1,c_2>0$ with $c_1x\log x<\sum_{k\le x}d(f(k))<c_2x\log x$ for $x\ge2$.

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