The present paper is devoted to the theory of discontinuous Markoff processes, that is processes where the transitions between states take place either by “jumps” of some specified kind, or by other means. States are taken as point$x$in an abstract space;$phases$are points ($x, t$) in the product state×time space; sets of states are denoted by$X$, sets of phases by$S$.

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In two earlier papers with the same title, we studied connections between Voronoi’s formula in the divisor problem and Atkinson’s formula for the mean square of Riemann’s zeta-function. Now we consider this correspondence in terms of segments of sums appearing in these formulae and show that a certain arithmetic conjecture concerning the divisor function implies best possible bounds for the classical error terms Δ($x$) and$E$($T$).