23 problems

Let $P^+(n)$ denote the largest prime factor of $n$. For $a,b\in[0,1]$, define … Let $\rho(t)$ denote the Dickman–de Bruijn function. Erdős–Pomerance conjecture. For every…

Uniform smooth-number cancellation conjecture. The paper conjectures that its main estimate should hold uniformly for every $2\leqslant y\leqslant x$:

Let $a$ be the fixed shift appearing in the statement, let $P^{+}(m)$ denote the largest prime factor of $m$, and let $\boldsymbol{\vartheta}$ and $\boldsymbol{\Theta}$ be real num…

Let $n>0$ be an integer and let $x\in\{0,1\}^n$ be a vector of bits of length $n$. For each $1\leq k\leq n$, let $q_k$ denote the $k$th odd prime. Prime smoothness and prescribed v…

Let $F(x)\in\mathbb{Z}[x]$, and let $\Psi_F^*(x,y)$ denote the counting function for the relevant integers whose values under $F$ satisfy the smoothness condition encoded by the so…

For $X,Y\to\infty$ with $X\geq Y$, define … and let … where $(\Delta/p)$ is the Legendre symbol and $\Psi(X,Y)$ counts the $Y$-smooth integers at most $X$. Weak Granville conjectur…

For $X,Y\geq1$, let $P(n)$ be the greatest prime factor of $n$, and define … Let $\Psi(X,Y)$ denote the count of positive integers at most $X$ whose prime factors are all at most…

Gordon–Pomerance conjecture. With the same hypotheses as Lemma Gordon–Pomerance, the displayed upper bound holds. The conjecture is used to obtain the conditional upper bound for t…

Granville–Soundararajan conjecture. There exists a constant $A>0$ such that, for every non-principal character $\chi$ modulo $q$ and every $1\leq x\leq q$, uniformly,

Smooth approximation conjecture. There exists a constant $A>0$ such that, uniformly for $1\leqslant x\leqslant T$ and $2T\leqslant t\leqslant5T$,

Smooth-number approximation conjecture. There exists a constant $A>0$ such that, for any $1\le x\le T$ and $2T\le t\le 5T$, uniformly as $T\to\infty$,

Soundararajan's conjecture. For any fixed value of $A>0$, if $q$ is sufficiently large, depending only on $A$, with $q\le y^A$ and $(a,q)=1$, then

Smooth numbers conjecture. If $y$ and $q$ are large with $q\leq y^A$, then, as $\frac{\log x}{\log y}\rightarrow\infty$,

Asymptotic conjecture for $M(p)$. As $p$ tends to infinity,

For a polynomial $P\in{\mathbb Z}[x]$, let $\Psi_P(x,y)$ count the integers among $\{P(n):1\leq n<x\}$ whose prime factors are all at most $y$. Weak smooth-values conjecture. For e…

Let $P\in{\mathbb Z}[x]$ be an irreducible polynomial of degree $d$, and let $\Psi_P(x,y)$ be the number of integers in the set $\{P(n):1\leq n<x\}$ whose prime factors are all at…

Weak XYZ conjecture. For every $\delta>0$, there are only finitely many solutions $(A,B,C)\in\frak X$ for which

XYZ conjecture (strong form). For every $\epsilon>0$, both of the following hold:

Fix $0<\epsilon<1$. An integer is $y$-smooth if all of its prime factors are at most $y$, and let $\mathcal{S}_{n^{\epsilon}}$ denote the set of integers $n$ with no prime factor g…

For real $x,y$, let $\Psi(x,y)$ denote the number of positive integers at most $x$ all of whose prime factors are at most $y$. Such integers are called $y$-smooth numbers. Let…

Relative-density conjecture for primitive solutions.

Dąbrowski's conjecture. The equation has exactly $28$ solutions $(x;\alpha_1,\ldots,\alpha_k)$ in positive integers, namely the solutions listed in the statement: $(3;3)$;…

Let $a_1,a_2,\ldots$ be independent uniform random integers in $\{1,\ldots,x\}$, and let $T$ be the least $t$ such that some subsequence of $a_1,\ldots,a_t$ has product equal to a…