12 problems

Cassaigne et al.'s conjecture. The values $\lambda(f(n))$ change sign infinitely often.

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 $\lambda$ be the Liouville function. Let $f(x)\in\mathbb Z[x]$ be a polynomial that is not of the form $cg(x)^2$, where $c\in\mathbb Z$ and $g(x)\in\mathbb Z[x]$. Chowla's poly…

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 $\bm{\lambda}(n)$ 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…

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

Let $f\in\mathbb{Z}[x_1,\ldots,x_h]$ be a primitive squarefree polynomial, let $\mathcal{A}$ be an arithmetic progression, and let $\mathcal{A}(X)$ denote its points of norm at mos…

Let $k\geq 1$, let $a_1,\dots,a_k$ be natural numbers, and let $b_1,\dots,b_k$ be distinct nonnegative integers satisfying … for $1\leq i<j\leq k$. Let $\lambda$ denote the Liouvil…

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

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.