61 problems

Let $\lambda$ denote the Liouville function, and let $H=\{h_1,\ldots,h_d\}$ be a non-empty set of non-negative integers. Chowla's conjecture. The correlation of the Liouville funct…

Conjectured optimal bound. For $1/\sqrt e\leqslant \alpha\leqslant 1$, one has

Sun's conjecture. For all $x\geq 3078$,

For integers $k\geq 2$ and $q\geq 1$, define the error term $E_{k,q}(x)$ by … where … Here $\varphi(q)$ is Euler's totient function, $\theta(q)$ is the number of positive squarefre…

Goldmakher's conjecture. Let $f\in\mathcal{F}$. Then for any $1\leq y\leq x$,

Let $s\geq 1$ and let $(\epsilon_1,\ldots,\epsilon_s)\in\{-1,1\}^s$ be a sign pattern. Chowla's sign-pattern conjecture. The set … has natural density $2^{-s}$. This is the sign-pa…

Let $x$ be sufficiently large. Consider a real-valued completely multiplicative function $f$ satisfying $|f(n)|\leq 1$ for every positive integer $n$, and suppose that $f(p)=0$ for…

Polynomial exponential-sum conjecture. If $g$ and $h$ satisfy

Pilatte's conjecture. These sums are bounded uniformly for all $x>0$ and all $\alpha\in E$ if and only if $f$ is eventually periodic. The conjecture was communicated to the authors…

Multiple recurrence conjecture. This liminf is positive. Furthermore, if all the rational polynomials have degree $0$ and $R_j(m_0,n_0)=1$ for $j=1,\ldots,\ell$ for some…

Mean convergence conjecture. These averages converge in $L^2(\mu)$ as $N\to\infty$. Furthermore, if all the rational polynomials have degree $0$, then the conclusion holds for all…

Let $\Omega(n)$ denote the total number of prime factors of $n$, counted with multiplicity, let $[N]=\{1,\ldots,N\}$, and for a finite set $S$ write…

Vanishing-correlation conjecture. If $P_1,P_2\in\mathbb{Z}[m,n]$ are irreducible binary quadratic forms that are not multiples of each other, then they are good for vanishing of co…

Let $k\geq 2$ be an integer, let $\mu^{(k)}$ denote the indicator function of the $k$-free integers, and let $h:\mathbb{N}\to\{-1,1\}$ be multiplicative. Aymone's conjecture. For e…

For real $a>0$, define … Let $\widetilde{W}_a(x)$ and $s_a$ be the normalized function and its limit superior defined in the source's preceding theorem. Large-parameter conjecture.…

For real $a>0$, let $\Omega(n)$ denote the number of prime factors of $n$, counting multiplicity, and define … For $a\geq 2$, the function has jumps of size $x^{\log a/\log 2}$ at…

Elliott-type correlation conjecture. If $P_1$ and $P_2$ are irreducible binary quadratic forms that are not multiples of each other, then they are good for vanishing of correlation…

Let $c\in\{\mu,\lambda\}$, where $\mu$ is the Möbius function and $\lambda$ is the Liouville function. Let $r$ be a prime modulus, let $\mathbb E_\chi$ denote the average over Diri…

Let $\mathcal{M}$ be the set of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$ with $|f(n)|\leq 1$. For $f,g\in\mathcal{M}$, define … A function is non-pretentious when it is…

Inhomogeneous-shift classification conjecture. If $\delta(\mathcal{N}_{f,a,b})\neq 0$ for some nonzero $a,b\in\mathbb{Z}$, then

Let $\mathbb{D}$ be the closed unit disc, and define the pretentious distance by … Let $k\ge1$ and let $h_1,\ldots,h_k\in\mathbb{N}$ be fixed and distinct. For $x\ge A\ge2$, let…

Let $\mathbb{D}$ be the closed unit disc, and call a sequence $a:\mathbb{N}\to\mathbb{D}$ deterministic if it is generated by a continuous observable on a zero-entropy topological…

Let $\mathbb{D}$ be the closed unit disc, let $\mathbb{D}(g_1,g_2;x)$ denote the pretentious distance, and let $\delta^+_{\log}$ denote upper logarithmic density. Let $k\ge1$, and…

Let $f:\mathbb{N}\to\{-1,0,+1\}$ be a non-pretentious multiplicative function. Write $1_{f(n)=0}$ for the multiplicative function that equals $1$ when $f(n)=0$ and $0$ otherwise. C…

Let $F_{2,3}$ denote the set of completely multiplicative functions $f:\mathbb{N}\to\{-1,1\}$ whose maximal run of consecutive $+1$ values has length $3$. For primes…