Explicit upper bounds for the remainder term in the divisor problem

Abstract

We first report on computations made using the GP/PARI package that show that the error term ∆(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 $|\Delta(x)| \le 0.397 x^{1/2}$ when $x \ge 5 560$ and that $|\Delta(x)| \le 0.764 x^{1/3}\log x$ 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