We obtain reasonably tight upper and lower bounds on the sum n x ϕ (⌊x/n⌋), involving the Euler functions ϕ and the integer parts ⌊x/n⌋ of the reciprocals of integers.

We study sums of the shape n x f (⌊x/n⌋) where f is either the von Mangoldt function or the Dirichlet-Piltz divisor functions. We improve previous estimates when f = Λ and f = τ , and provide new results when f = τ r with r 3, breaking the 1 2 -barrier in each case. The functions f = µ 2 and f = 2 ω are also investigated.

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).

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