On some log-cosine integrals related to ζ(3), ζ(4), and ζ(6)

摘要

We evaluate a variety of log-cosine integrals, one of which has value a rational multiple of πζ(4), while others evaluate to the form π[aζ(6)+b(3/4)ζ2(3)], where a>0 is rational and b is 1 or 2, and ζ is the Riemann zeta function. We show that these results may be obtained starting from a known tabulated result. The derivation yields results for certain classes of logarithm-trigonometric integrals, and indicates how further analytic results on digamma series may be obtained.