Explicit upper bounds for the remainder term in the divisor problem

We first report on computations made using the GP/PARI package that show that the error term $\Delta(x)$ in the divisor problem is $=\mathscr{M}(x,4)+ O^*(0.35 x^{1/4}\log x)$ when $x$ ranges $[1 081 080, 10^{10}]$, where $\mathscr{M}(x,4)$ is a smooth approximation. The remaining part (and in fact most) of the paper is devoted to showing that $\vert\Delta(x)\vert\le 0.397 {x^{1/2}}$ when $x\ge 5 560$ and that $\vert\Delta(x)\vert\le 0.764 {x^{1/3}\log x}$ when $x\ge 9 995$. Several other bounds are also proposed. We use this results to get an improved upper bound for the class number of a quadractic imaginary field and to get a better remainder term for averages of multiplicative functions that are close to the divisor function. We finally formulate a positivity conjecture concerning $\Delta(x)$.