2 problems

Let $n\in\mathbb{N}$ with $n\geqslant 2$. Let $\Phi_n(t)$ be the truncated kernel considered above, let $\omega_n=(1/2)\log n$, and let $\tau_n$ denote its smallest positive zero.…

Let $\Xi_k(z)=\sum_{j=1}^{k}\Phi_j(z)$ and let $\Phi_{k+1}(z)$ be the next summand in the truncated Riemann Xi-function series. For $k\geq1$ and $0\leq t\leq1$, consider…