In [Er79] asks the stronger version with $2$ replaced by any constant $c>1$.

The answer is yes (also to this stronger version), proved by Maier and Tenenbaum [MaTe84]. (Tenenbaum has told me that they received \$650 for their solution.)

See also [449].

Conjectured by Erdős and Pomerance [ErPo78], who proved that this set and its complement both have positive upper density.

In [Er79e] Erdős also asks whether, for every $\alpha>0$, the density of the set of $n$ where \[P(n+1)>P(n)n^\alpha\] exists.

The sequence of such $n$ is A070089 in the OEIS.

Pillai [Pi29] proved $V(x)=o(x)$. Erdős [Er35b] proved $V(x)=x(\log x)^{-1+o(1)}$.

The behaviour of $V(x)$ is now almost completely understood. Maier and Pomerance [MaPo88] proved \[V(x)=\frac{x}{\log x}e^{(C+o(1))(\log\log\log x)^2},\] for some explicit constant $C>0$. Ford [Fo98] improved this to \[V(x)\asymp\frac{x}{\log x}e^{C_1(\log\log\log x-\log\log\log\log x)^2+C_2\log\log\log x-C_3\log\log\log\log x}\] for some explicit constants $C_1,C_2,C_3>0$. Unfortunately this falls just short of an asymptotic formula for $V(x)$ and determining whether $V(2x)/V(x)\to 2$.

In [Er79e] Erdős asks further to estimate the number of $n\leq x$ such that the smallest solution to $\phi(m)=n$ satisfies $kx<m\leq (k+1)x$.

See also [417] and [821].

It is trivial that $V'(x) \leq V(x)$. See also [416].

Besicovitch [Be34] proved that $\liminf \delta(n)=0$. Erdős [Er35] proved that $\delta(n)=o(1)$. Erdős [Er60] proved that $\delta(n)=(\log n)^{-\alpha+o(1)}$ where \[\alpha=1-\frac{1+\log\log 2}{\log 2}=0.08607\cdots.\] This estimate was refined by Tenenbaum [Te84], and the true growth rate of $\delta(n)$ was determined by Ford [Fo08] who proved \[\delta(n)\asymp \frac{1}{(\log n)^\alpha(\log\log n)^{3/2}}.\]

Among many other results in [Fo08], Ford also proves that the second conjecture is false, and more generally that if $\delta_r(n)$ is the density of integers with exactly $r$ divisors in $(n,2n)$ then $\delta_r(n)\gg_r\delta(n)$.

See also [448], [692], and [693].

This is false, and was disproved by Erdős and Tenenbaum [ErTe81], who showed that in fact the upper density of the set of such $n$ is $\asymp \epsilon^{1-o(1)}$ (where the $o(1)$ in the exponent $\to 0$ as $\epsilon \to 0$).

A more precise result was proved by Hall and Tenenbaum [HaTe88] (see Section 4.6), who showed that the upper density is $\ll\epsilon \log(2/\epsilon)$. Hall and Tenenbaum further prove that $\tau^+(n)/\tau(n)$ has a distribution function.

Erdős and Graham also asked whether there is a good inequality known for $\sum_{n\leq x}\tau^+(n)$. This was provided by Ford [Fo08] who proved \[\sum_{n\leq x}\tau^+(n)\asymp x\frac{(\log x)^{1-\alpha}}{(\log\log x)^{3/2}}\] where \[\alpha=1-\frac{1+\log\log 2}{\log 2}=0.08607\cdots.\]

See also [446] and [449].

Linnik's theorem implies that $p_n\leq n^{O(1)}$. Erdős [Er79e] writes it is 'easy to show' that for infinitely many $n$ we have $p_n <m_n$.

Erdős writes it is 'easy' to prove $\frac{1}{X}\sum_{n\leq X}G(n)\to \infty$.

Terence Tao has observed that, for any divisor $m\mid n$, \[\frac{\tau(n/m)}{m} \leq G(n) \leq \tau(n),\] and hence for example $\tau(n)/4\leq G(n)\leq \tau(n)$ for even $n$. It is easy to then see that $G(n)$ grows on average, and in general behaves very similarly to $\tau(n)$ (and in particular the answer to the first question is yes). Tao suggests that this was a mistaken conjecture of Erdős, which he soon corrected a year later to [448].

Erdős believes that this is not possible, but could not disprove it. He could show that $p_k$ is about $e^{e^k}$ for almost all $n$, but the maximal value of $d_k(p)$ is assumed for much smaller values of $p$, at \[p=e^{(1+o(1))k}.\]

A similar question can be asked if we consider the density of integers whose $k$th smallest divisor is $d$. Erdős could show that this function is not unimodular.

If $A$ is a set of prime numbers then a necessary and sufficient condition is that $\sum_{p\in A}\frac{1}{p}=\infty$.

The general situation is more complicated. For example suppose $A$ is the union of $(n_k,(1+\eta_k)n_k)\cap \mathbb{Z}$ where $1\leq n_1<n_2<\cdots$ is a lacunary sequence. If $\sum \eta_k<\infty$ then the density of $M_A$ exists and is $<1$. If $\eta_k=1/k$, so $\sum \eta_k=\infty$, then the density exists and is $<1$.

Erdős writes it 'seems certain' that there is some threshold $\alpha\in (0,1)$ such that, if $\eta_k=k^{-\beta}$, then the density of $M_A$ is $1$ if $\beta <\alpha$ and the density is $<1$ if $\beta >\alpha$.

Erdős proved that \[\delta_1(n,m) \ll \frac{1}{(\log n)^c}\] for all $m$, for some constant $c>0$. Sharper bounds (for various ranges of $n$ and $m$) were given by Ford [Fo08].

See also [446].

See also [446].

Carmichael has asked whether there is an integer $n$ for which $\phi(m)=n$ has exactly one solution, that is, $\frac{f_{\max}(n)}{f_{\min}(n)}=1$. Erdős has proved that if such an $n$ exists then there must be infinitely many such $n$.

See also [51].

Linnik's theorem implies that there exists such a sequence of primes with \[q_k \leq e^{e^{O(k)}}.\]

See also [696].

Erdős writes it is 'easy to see' that $h(n)\to \infty$ for almost all $n$, and believed he could show that the normal order of $h(n)$ is $\log_*(n)$ (the iterated logarithm).

See also [695].

It is trivial that $\delta(m,\alpha)\to 0$ if $\alpha <1$, and Erdős could prove that the same is true for $\alpha=1$.

See also [696].