Open-problem statements, with their sources attached.

Let $f$ be the asymptotic distribution function of $\varphi(n)/n$, so that for each $c \in [0,1]$, $f(c)$ is the natural density of $\{n : \varphi(n) < cn\}$. Is it true that there is no $x$ such that the derivative $f'(x)$ exists and is positive?

Is there an infinite set $A \subset \mathbb{N}$ such that for every $a \in A$, there is an integer n such that $\phi(n)=a$, and yet if $n_a$ is the smallest such integer, then $\frac{n_a}{a} → \infty$ as $a → ∞$?

Chowla's cosine problem*

If $A\subset \mathbb{N}$ is a finite set of positive integers of size $N > 0$ then is there some absolute constant $c>0$ and $\theta$ such that $\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?$

Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$ $\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?$

Let $f$ be a Rademacher multiplicative function. Does there exist some constant $c > 0$ such that, almost surely,

Let $\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\subseteq \{1,\ldots,N\}$ has size at least $\epsilon N$ then there must be distinct $a,b,c\in A$ such that $[a, b]=[b, c]=[a, c],$ where $[\cdot, \cdot]$ denotes the least common multiple?

Let $\tau(n)$ count the number of divisors of $n$. Is there some $n > 24$ such that

Erdős says 'it is extremely doubtful' that there are infinitely many such $n$, and in fact suggests that

Erdős says it 'seems certain' that for every $k$ there are infinitely many $n$ for which

Is there and $A \subset \mathbb{N}$ is such that $\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}$ exists and is $\ne 0$?

Can the product of an arithmetic progression of positive integers $n, n + d, ..., n + (k - 1)d$ of length ≥ 4, with $(n, d) = 1$, be a perfect power?

Denote by $M(n, k)$ the least common multiple of the finite set $\{n+1, \dotsc, n+k\}$. Is it true that for all $m \geq n + k$, we get $M(m, k) \neq M(n, k)$?

Is $\sum_{n=2}^\infty \frac{1}{n!-1}$ irrational?

Is it true that, for all sufficiently large $n$, there exists some $k$ such that

where $p(m)$ denotes the least prime factor of $m$?

Can one prove this is false if we replace $k^2+1$ by $e^{(1+\epsilon)\sqrt{k}}+C_\epsilon$, for all $\epsilon>0$, where $C_\epsilon>0$ is some constant?

Erdős problem 681.* Is it true that for all large $n$ there exists $k$ such that $n + k$ is composite and $p(n+k) > k^2$, where $p(m)$ is the least prime factor of $m$ ?

There exists $c > 0$ such that $P(n, k) > \min\{n-k+1, k^{1 + c}\}$ for all $0 < k < n$.}

Standard heuristics suggest that $P(n, k) > e^{c\sqrt{k}}$ for some constant $c > 0$.

Can every integer $N≥2$ be written as $N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$ for some $k≥2$ and $m≥n+k$?

Can every square $N≥2$ be written as $N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$ for some $k≥2$ and $m≥n+k$?