2 problems

For $j\in\mathbb N_0$, let $B_{2j+1}^\sim(z)$ be the Hilbert-transformed Bernoulli function, let $B_m(z)$ denote the Bernoulli polynomials, let $\eta(s)$ be the Dirichlet eta funct…

Let $\eta(s)=\sum_{n=1}^{\infty}(-1)^{n-1}n^{-s}$ be the Dirichlet eta function, and let $\alpha$ and $t$ be real parameters. Consider the ratio of its values at symmetric points a…