Abstract. 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 * (0.35 x 1/4 log x) when x ranges [1 081 080, 10 10 ], 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)| ≤ 0.397 x 1/2 when x ≥ 5 560 and that |Δ(x)| ≤ 0.764 x 1/3 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).

Let π (x) be the number of primes not exceeding x. We produce new explicit bounds for π(x) and we use them to obtain a fine frame for the remainder term in the asymptotic formula of the sum 2≤n≤x 1/π(n). Mathematics Subject Classification. 11A41, 11Y60, 26D15, 40A25.

A generalized formula is obtained for the sum of inverses of the prime counting function for a large class of arithmetical functions related to the iterated logarithms.

Robin’s criterion says that the Riemann Hypothesis is equivalent to \[\forall n\geq 5041, \ \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n,\] where \sigma(n) is the sum of the divisors of n, \gamma represents the Euler–Mascheroni constant, and \log_i denotes the i-fold iterated logarithm. In this note we get the following better effective estimates: \begin{equation*} \forall n\geq3, \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n+\frac{0.3741}{\log_2^2n}. \end{equation*} The idea employed will lead us to a possible new reformulation of the Riemann Hypothesis in terms of arithmetic functions.