  The Landau-Khalatnikov-Fradkin (LKF) transformation is a powerful and elegant
transformation allowing to study the gauge dependence of the propagator of
charged particles interacting with gauge fields. With the help of this
transformation, we derive a non-perturbative identity between massless
propagators in two different gauges. From this identity, we find that the
corresponding perturbative series can be exactly expressed in terms of a hatted
transcendental basis that eliminates all even Euler $\zeta$-functions. This
explains the mystery of even $\zeta$-values observed in multi-loop calculations
of Euclidean massless correlators for almost three decades now. Our
construction further allows us to derive an exact formula relating hatted and
standard $\zeta$-functions to all orders of perturbation theory.
