60 problems

Let $x$ and $y$ be positive real numbers, and consider intervals of length $y$ whose location is near $x$. Let the maximum be taken over such intervals of the number of primes they…

Square-prime short-interval conjecture. For all $x \geqslant 2$,

Let $x\geq 1$, let $H$ be a real number, and let $\varepsilon\in(0,1/100)$. The function $\mu^2(n)$ is the indicator of squarefree integers. Short-interval conjecture. Uniformly fo…

Let $\mu(n)$ denote the Möbius function. For $X\to\infty$, let $H=X^\delta$ with $\delta\in(0,1)$ and average over $x\in[X,2X]$. Good-Churchhouse conjecture. … This conjectures tha…

Let … For any $0<\varepsilon<1/4$ and $x^\varepsilon\leqslant U\leqslant x^{1/2-\varepsilon}$, Jutila's short-interval conjecture. … This predicts square-root cancellation in the d…

Let $\Delta(x)$ be the Dirichlet divisor error term, and define, for $t\geq2$, … where $\gamma$ is Euler's constant. Here $T$ is large and $U$ satisfies $1\ll U\ll T^{1/2}$. Jutila…

Hua's conjecture. Every sufficiently large $n\in\mathcal H_2$ can be represented as the sum of a prime and the square of another prime, and every sufficiently large…

For $x\geq 1$ and $h\geq 1$, define … where $\mu(n)$ is the Möbius function. Squarefree numbers in short intervals conjecture. For every $\varepsilon>0$, there exists…

Let $X$ be large, let $\delta$ be a positive real parameter, and let $\psi(x)=\sum_{n\leq x}\Lambda(n)$ be Chebyshev's function. For fixed $\epsilon>0$ and $\lambda>0$, define…

Let $X$ be large, let $h$ be a positive real parameter, and let $\psi(x)=\sum_{n\leq x}\Lambda(n)$ be Chebyshev's function. For fixed $\epsilon>0$ and $\lambda>0$, define…

For a real number $\theta>0$ and sufficiently large $x$, the interval $[x,x+x^\theta]$ contains a prime number. The short-interval prime conjecture. Given $\theta>0$, for all suffi…

For a positive real number $\alpha$, let $\lambda(\alpha)$ be the smallest non-negative real such that, for every $\varepsilon>0$ and all sufficiently large $n$, the Lebesgue measu…

Let $\pi(t)$ denote the number of primes at most $t$. For a real parameter $\lambda=(\log x)^{o(1)}$ and an integer $k\ll (\log_2 x)^2$, count integers $n\leqslant x$ according to…

Let $p_n$ denote the $n$th prime. For $varepsilon>0$ and $0\leqslant \lambda\leqslant (\log x)^{1-varepsilon}$, consider the number of primes $p_n\leqslant x$ whose following prime…

Let $L(s,\pi)$ be a primitive automorphic $L$-function of degree $d>1$, and let … where $m_\pi=1$ for the Riemann zeta function and $m_\pi=0$ otherwise. Here…

Let $\lambda:\mathbb{N}\to\{+1,-1\}$ be the Liouville function, defined by $\lambda(n)=(-1)^{\Omega(n)}$, where $\Omega(n)$ counts prime factors with multiplicity. Let $H=H(X)$ be…

Let $\omega(n)$ denote the number of distinct prime factors and define … where $\log_2 n=\log\log n$ and $\log_3 n=\log\log\log n$. Short-interval prime-factor conjecture. For some…

For $x>0$, define … where $\log_2 n=\log\log n$. Short-interval conjecture. For $\varepsilon>0$, there is a constant $C$ such that, for sufficiently large $x$, … and … This conject…

Let $wp:Z\toZ$ be a bracket polynomial of complexity $delta$, and let $C>0$, $epsilon>0$. Theorem gives bounds in short intervals when $wp$ is an ordinary polynomial, with implied…

Conjectured square-full short-interval error term. The error term in the asymptotic formula for the number of square-full integers in $(x,x+H]$ is conjectured to be of order…

Let $n\geq 1$ be an integer, let $\mathcal{D}_n$ be the set of positive divisors of $n$, and define … Fix real numbers $\theta$ and $\epsilon$ satisfying $0<\epsilon<\theta<1$. Sho…

Variance conjecture.

Extended tail-bound conjecture. As $x\to\infty$,

The short-interval k-free integer conjecture. Uniformly for $x^{\epsilon}\leq H\leq x$,

Let $k\ge 2$ be an integer, let $x\ge 1$ be real, and let $Q_k(t)$ denote the number of $k$-full numbers at most $t$. A number is $k$-full if every exponent in its prime factorizat…