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).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.