Some integrals of the Dedekind $\eta$ function

Abstract

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 $${\int }_0^{\infty }{\eta }^6(ix)dx={\int }_0^{\infty }x{\eta }^6(ix)dx=\frac{1}{8\pi }{\left(\frac{\Gamma (1/4)}{\Gamma (3/4)}\right)}^2,$$ 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 ${\gamma }_1(a)$.

21 pages, no figures