Extended higher Herglotz function II

Very recently, Radchenko and Zagier revived the theory of Herglotz functions. The main goal of the article is to show that one of the formulas on page 220 of Ramanujan's Lost Notebook actually lives in the realms of this theory. As a consequence of our general theorem, we derive an interesting identity analogous to Ramanujan's formula for ζ(2m+1). We also introduce a character analogue of the Herglotz function and initiate its theory by obtaining an elegant functional equation governed by it.