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 Δ(x) in the divisor problem is = M (x, 4)+ O∗(0.35x1/4 log x) when x ranges [1 081 080, 1010], where M (x, 4) is a smooth approximation. The remaining part (and in fact most) of the paper is devoted to showing that |Δ(x)| ≤ 0.397x1/2 when x ≥ 5 560 and that |Δ(x)| ≤ 0.764x1/3 log x when x ≥ 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 Δ(x).