6 problems

Let $\rho=\beta+i\u03b3$ and $\rho'=\beta'+i\u03b3'$ denote non-trivial zeros of $\zeta(s)$ and $\zeta'(s)$, respectively, and assume the Riemann Hypothesis. Let $\gamma^+$ be the…

Let $N_0(T)$ denote the number of zeros of the function $\zeta\left(\frac{1}{2}+it\right)$ with $t\in(0,T)$. Selberg established that, for every $\epsilon>0$, … for…

Let $\zeta_N(s;R)=\sum_{n=1}^N B_{R(n)}(s)$ and $\widetilde{\zeta}_N(s;\widetilde R)=\sum_{n=1}^N\widetilde B_{\widetilde R(n)}(s)$ be the re-arranged sections defined by the sourc…

Let $B_n(s)=\frac12[n^{-s}+\chi(s)n^{s-1}]$ be the summands in the approximate functional equation, and let $\zeta_{\lfloor t/2\rfloor}(s)$ be formed from the first…

For $N\in\mathbb{N}$, let … In the region … these sections approximate the Riemann zeta function $\zeta(s)$. Spira's conjecture. In this region, $\zeta_N(s)$ has zeros only on the…

GUE conjecture. Assume the Riemann hypothesis. For every fixed $k$ and every such test function $\eta$,