58 problems

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,

Let $A>0$ be a fixed real number, and let $y$ and $q$ be large positive numbers satisfying $q\leq y^A$. For $x$ tending to infinity with … let $\Psi(x,y;q,a)$ count the $y$-smooth…

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…

Booker–Pomerance conjecture. The quantity $M(p)$ satisfies

For an arithmetic function $f$, define … Let $\chi_0$ denote the principal character modulo $q$. Smooth-number approximation conjecture. There exists a constant $A>0$ such that for…

Smooth-number distribution conjecture.

Let $k$ be the fixed degree, let ${\Cal A}(P,R)$ be the set of natural numbers not exceeding $P$ whose prime divisors are at most $R$, and define … For a positive integer $t$, let…

Let $n$ be a positive integer, and let $Y$ and $Y+1$ be consecutive integers satisfying $up(Y)+ up(Y+1)=nup(n!)$ with the full valuation $nup(n!)$ assigned exclusively to one of t…

Let $f:bb N\to\mathbb U$ be a multiplicative function, let $\alpha\in[0,1)$ be approximated by a reduced fraction $a/q$ with $q\leqslant x$ and $|\alpha-a/q|\leqslant 1/(qx)$, and…

Let $\{N+1,\dots,N+H\}$ be a bad interval, and let $x$ be a large parameter with the interval under consideration lying in the relevant range up to $x$. Erdős–Graham bad-interval c…

Short-interval smoothness conjecture. For any $epsilon>0$, there is a positive number $d>0$ such that the interval

Let $A=(a_1,a_2,\ldots)$ be the increasing sequence of $3$-smooth integers, so each $a_i=2^{x_i}3^{y_i}$ for non-negative integers $x_i,y_i$. A sum of distinct elements of $A$ has…

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

Let $P(n)$ denote the largest prime factor of $n$, let $f$ be a Steinhaus random multiplicative function, and fix $A>0$ and a sufficiently small $\varepsilon>0$. For $T>0$, exclude…

For $x\ge y\ge 1$, let … where $P^+(n)$ is the greatest prime factor of $n$, with $P^+(1)=1$, and let … while $\pi(x)$ denotes the number of primes at most $x$. Thus $\Psi(x,y)/x$…

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 $p$ range over primes, let $a$ be the fixed shift appearing in the statement, and let $P^{+}(m)$ denote the largest prime factor of $m$. The shifted-prime largest-factor distri…

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…

Asymptotic estimates conjecture. For all $a>0$, the estimates

Prime-divisor counting conjecture. For all primes $\ell$ and $1<\alpha\leq2$,

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…

Let $x$ be a positive integer, let $B=L(\alpha,x):=\exp(\alpha\sqrt{\log x\log\log x})$ with $\alpha>0$, and let $s_B(m)$ denote the largest $B$-smooth divisor of $m$. Define … and…

Let $f(t)\in\mathbb{Z}[t]$ have distinct irreducible factors over $\mathbb{Z}[t]$ of degrees $d_1,\ldots,d_g\geq 1$. Let $\Psi_f(x,y)$ denote the number of integers $n\in[1,x]$ suc…

Concentration and limiting-distribution conjecture. The measure $\mu_{n,m}$ concentrates around its mean and, up to a suitable normalization constant, admits an asymptotic limiting…

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…