In this paper, we shall investigate several properties of integrals defined by $\int_1^{\infty}t^{-\theta}P(t)\log^jtdt$ with a complex variable $\theta$ and a non-negative integer $j$, where $P(x)$ is the error term in the circle problem of Gauss. We shall also study the analytic continuation of several types of the Dirichlet series related with the circle problem, and study a proof of the functional equation of the Dedekind zeta-function associated with the Gaussian number field ${\mathbb{Q}}(\sqrt{-1})$.