4 problems

Sum rule conjecture. For each pair of integers $(n,k)$ with $n\geq 2k\geq 2$, the following combination of Bessel moments vanishes:

Assume $k$ is even, and let $w”$ be the unit-modulus local factor appearing in the epsilon factor at $2$. Broadhurst50Roberts' local root-number conjecture. One has … This conjectu…

Broadhurst50Roberts' local conductor conjecture. Based on numerical data, one has

Let $k$ be even. In the completed even symmetric-power $L$-function, let $\varepsilon_k$ be the sign in the functional equation, and define $t_k$ by … where the sums run over prime…