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).
"Riemann’s zeta-function and the divisor problem. III." Ark. Mat. 53 (2) 303 - 315, October 2015. https://doi.org/10.1007/s11512-014-0204-9