Explicit upper bounds for the remainder term in the divisor problem

Abstract

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).

Explicit estimates on the summatory functions of the Möbius function with coprimality restrictions

“…This process is also well detailed in [2]. When the model is f 0 = 1, the situation is readily cleared out; it is also well studied when this model is the divisor function in [1,Corollary 2.2]. We signal here that the case of the characteristic function of the squarefree numbers is specifically handled in [4].…”

“…The computations have been run with PARI/GP (see [11]), speeded by using gp2c as described for instance in [1]. We mention here that [6] proposes an algorithm to compute isolated values of M (x).…”

Two explicit divisor sums

“…The rolled-gold example is n≤x d(n), which is gives a bound of the form (2) for k = 2. Berkané, Bordellès and Ramaré [1,Thm. 1.1] gave several pairs of values (α, x 0 ) such that n≤x d(n) = x(log x + 2γ − 1) + ∆(x), (4) holds with |∆(x)| ≤ αx 1/2 for x ≥ x 0 , and where γ is Euler's constant.…”

“…It seems hopeless to give a bound on the implied constant in this estimate. Therefore weaker 1 , yet-still-explicit bounds such as those in [1] are very useful in applications.…”

Explicit estimates on several summatory functions involving the Moebius function

“…OLIVIER RAMARÉ Scripts. All the computations used have been achieved via GP/PARI (see [18]) often sped up by using gp2c as described for instance in [5]. To give a flavor, we have used for instance the command gp-run -g AsymptoticBoundsFor M.gp.…”