Theory of log-rational integrals

Herein we consider the functional L[f] := integral(1)(0) f(u)log u du, especially when f is a rational polynomial, in which case we refer to L as a "log-rational integral." The relevance of the present study runs like so: A decade ago, a mysterious class of conjectured Clausen identities ("resonances") was uncovered experimentally by J. Borwein and D. Broadhurst via the powerful numerical techniques of D. Bailey and others. Most of said conjectures remain unproven. Herein we show that every such Clausen conjecture can be put in an equivalent "log-rational" form L[r] = 0, where r is an explicit rational polynomial. Remarkably, the conjectured resonances can be interpreted as hypotheses regarding the zeros of certain Hurwitzzeta superpositions. It is hoped that these various equivalencies will lead to ultimate resolution of such experimentally motivated, highly nontrivial conjectures.