7 problems

Williamson's conjecture. There exists $\alpha'$ with $2<\alpha'<3$ such that $\Psi_\alpha(x)$ remains nonnegative for all $x>0$ when $\alpha\geq\alpha'$, but changes sign for…

For a prime discriminant $p$, define the quadratic character sum … Let the relative proportion be taken over prime discriminants $p\leq x$ for which $S_{\chi_p}(N)\geq0$ for all…

Let $\{A(m,1)\}_{m\geq 1}$ be the sequence of Hecke eigenvalues of a self-dual Hecke–Maass cusp form, and let $\#\mathcal X$ denote the cardinality of a set $\mathcal X$. Positive-…

Let $f$ be the density function of a positive self-decomposable distribution, with Lévy measure having density $\nu$ on $(0,\infty)$. Suppose that the limit … exists and is greater…

Erdős–Pólya–Turán conjecture. The expression $\sum_{i=1}^{\ell}\alpha_i d_{n+i}$ changes sign infinitely often as $n$ runs through all integers if and only if the nonzero elements…

Erdős–Pólya–Turán conjecture. The sequence $\sum_{i=1}^k a_i p_{n+i}$ has infinitely many sign changes if and only if the nonzero elements among $\alpha_1,\dots,\alpha_k$ do not al…

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…