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Motivated by their role for integrality and integrability in topologicalstring theory, we introduce the general mathematical notion of "s-functions" asintegral linear combinations of poly-logarithms. 2-functions arise as diskamplitudes in Calabi-Yau D-brane backgrounds and form the simplest and mostimportant special class. We describe s-functions in terms of the action of theFrobenius endomorphism on formal power series and use this description tocharacterize 2-functions in terms of algebraic K-theory of the completed powerseries ring. This characterization leads to a general proof of integrality ofthe framing transformation, via a certain orthogonality relation in K-theory.We comment on a variety of possible applications. We here consider only powerseries with rational coefficients; the general situation when the coefficientsbelong to an arbitrary algebraic number field is treated in a companion paper.

Framing the Di-Logarithm (over Z)