Let $\eta$ be the weight $1/2$ Dedekind function. A unification and generalization of the integrals $\int_0^\infty f(x)\eta^n(ix)dx$, $n=1,3$, of Glasser \cite{ glasser2009} is presented. Simple integral inequalities as well as some $n=2$, $4$, $6$, $8$, $9$, and $14$ examples are also given. A prominent result is that $$\int_0^\infty \eta^6 (ix)dx= \int_0^\infty x\eta^6 (ix)dx ={ 1 \over { 8\pi}}\left({ { \Gamma(1/4)} \over { \Gamma(3/4)}}\right)^2,$$ where $\Gamma$ is the Gamma function. The integral $\int_0^1 x^{ -1} \ln x ~\eta(ix)dx$ is evaluated in terms of a reducible difference of pairs of the first Stieltjes constant $\gamma_1(a)$.
Categories: math.NT