---

Statement

Setting (p. 1). $N_f^+(x)$ is the set of $n\le x$ with $n=k+f(k)$ for some $k$, and $\tau(n)=\sum_{d\mid n}1$ is the divisor function.

Theorem 1.2 (p. 2, quoted). "$x\ll N_\tau^+(x)\leqslant 0.94x$."

So the integers of the form $k+\tau(k)$ have positive lower density and upper density at most $0.94$. The upper bound comes from the uneven distribution of $k+\tau(k)$ modulo $3$ (p. 2): the paper proves

$$\#\{k\le x:k+\tau(k)\equiv0\ (\mathrm{mod}\ 3)\} =\Bigl(\frac13+\frac{\zeta(3)}{12\zeta(2)}+o(1)\Bigr)x ,$$

whose constant is $0.394\ldots$ (p. 8, (3.6)). The paper reports numerical calculations predicting $N_\tau^+(x)\approx0.67x$ (p. 1); that is not a result.

Proof pointer

Section 3, pp. 6--10. Lower bound (§3.1, pp. 6--7): the argument of Theorem 1.1 with $l$ odd and squarefree, so that $\tau(lp)=2^{\omega(l)+1}$; the off-diagonal energy is now controlled by Luca and Shparlinski's bound for the mean of $(\sigma(2^r-1)/(2^r-1))^2$. Upper bound (§3.2, pp. 8--10): granted (3.6), the $(1-c+o(1))x$ integers $k\le x$ with $k+\tau(k)\not\equiv0$ (mod 3) cannot cover the $2x/3+O(1)$ integers $n\le x$ that are $\pm1$ (mod 3), which leaves at least $(\zeta(3)/(12\zeta(2))-o(1))x\ge0.06x$ of them unrepresented. (3.6) is proved by splitting by the residue of $k$ mod 3 and evaluating Dirichlet series; one of them is $L^{-1}(s,\chi_3)L(3s,\chi_3)$, whose coefficient sum is $\ll x\exp(-c\sqrt{\log x})$ by a lemma of Kucheriaviy (p. 10, (3.10)).

Read depth

Claims checked: the statement, (3.6) and the deduction of the upper bound from it were read on the print (arXiv v1, pp. 1--2, 6--10); the rest of Section 3 was followed for structure. Nothing here is independently reviewed.

Dependencies

The method of Theorem 1.1, for the lower bound. External inputs named by the paper: Selberg's sieve, Luca and Shparlinski's moment bound for $\sigma(2^r-1)/(2^r-1)$, Changa's Lemma 3.1 and Kucheriaviy's Lemma 10.

Source. M. R. Gabdullin, V. V. Iudelevich and F. Luca, Numbers of the form $k+f(k)$, J. Number Theory 262 (2024), 58--85, doi:10.1016/j.jnt.2024.03.010; arXiv:2306.16035. Labels and pages are those of the arXiv v1 edition named on the source card.

Bears on

None among the Erdős problems recorded here.