2 problems

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…

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…