22 problems

Let $f\in\mathbb Z[x]$ be a polynomial that does not have the form $c(g(x))^2$. Define … where $\Omega:\mathbb Q^\times\to\mathbb Z$ is the group homomorphism satisfying…

Let $\lambda$ be the Liouville function. Fix $k\geqslant1$ and distinct positive integers $h_1,\dots,h_k$. Logarithmic Chowla conjecture. One has … as $X\to\infty$. This logarithmi…

Corrádi–Kátai conjecture. The convolution satisfies

Let $\lambda(n)=(-1)^{\Omega(n)}$ be the Liouville function, where $\Omega(n)$ counts prime factors with multiplicity. Turán's conjecture. The partial sums … are always positive. T…

Two-variable Chowla conjecture. The sequence $\lambda(f(m,n))$ has average value $0$ as $(m,n)$ ranges over lattice points in any sector of the plane with vertex at the origin. Thi…

One-variable Chowla conjecture. The sequence $\lambda(f(n))$ has average value $0$ as $n$ ranges over any arithmetic progression. Equivalently, for every arithmetic progression…

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…

AIM workshop conjecture. For every integer $N\geq2$,

Let $\lambda$ be the Liouville function on the positive integers, and let $\chi_{-4}$ be the nontrivial quadratic character modulo $4$. Shanks' bias conjecture. There are more posi…

Let $\bm{\lambda}(n)$ be the Liouville function, and for $N\in\mathbb{N}$ define … For a fixed exponent $p$, let $\|L_N\|_p$ denote its $L^p$ norm on the unit circle.…

Let $\bm{\lambda}(n)$ be the Liouville function, defined by $\bm{\lambda}(1)=1$ and $\bm{\lambda}(n)=(-1)^r$ when $r$ is the number of prime factors of $n$ counted with multiplicit…

Liouville two-quadratic correlation problem. Prove that

Frantzikinakis's conjecture. For any such multiplicative functions, one has

Liouville at polynomials conjecture. If $P$ is non-square and $v\in\{-1,+1\}$, then $\lambda(P(n))=v$ for infinitely many $n\in\mathbb{N}$.

Let $s\geq 0$. Let $G/\Gamma$ be an $s$-step nilmanifold, where $G$ is a connected, simply connected nilpotent Lie group of step $s$ and $\Gamma$ is a cocompact discrete subgroup.…

For a finitely supported function $g\colon\mathbb{Z}\to\mathbb{C}$ and $d\geq 1$, define its Gowers norm by … where $|\vec\omega|=\omega_1+\dots+\omega_d$ and…

Let $\lambda(n)=(-1)^{\Omega(n)}$ be the Liouville function, where $\Omega(n)$ counts prime factors with multiplicity. Haselgrove's disproof of Pólya's and Turán's conjectures. Pól…

Refined half-weight conjecture. As $x$ tends to infinity,

Mossinghoff–Trudgian conjecture. The weighted sum $L_{1/2}(x)$ is nonpositive for all $x\geq 17$. This is motivated in the source by heuristic and computational evidence through…

Mossinghoff–Trudgian conjecture. For $0<\alpha<1/2$, the weighted sum $L_{\alpha}(x)$ changes sign infinitely often. The source notes that this has been proved unconditionally for…

Half-power sign conjecture.

Alpha-plus conjecture. For a fixed $\alpha$ with $0<\alpha<1$, one proposes that