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