Computing the divisor summatory function in $N^{9\mathrm{/}28+o(1)}$

We present an analytic algorithm to compute isolated values of the divisor summatory function in $N^{9\mathrm{/}28+o(1)}$ expected time, improving upon existing methods that require $O(N^{1\mathrm{/}3})$ time. Our method significantly departs from elementary approaches to analyze the geometry of the Dirichlet hyperbola, relying instead on Voronoï summation and accelerating the evaluation of the resulting exponential sum.