Values and recurrence relations for integrals of powers of arctan and logarithm and associated Euler-like sums

Abstract

In this paper, we give evaluations of integrals involving the arctan and the logarithm functions, and present several new summation identities for odd harmonic numbers and Milgram constants. These summation identities can be expressed as finite sums of special constants such as $\pi$, the Catalan constant, the values of Riemann zeta function at the positive odd numbers and $\ln2$ etc.. Some examples are detailed to illustrate the theorems.