Explicit estimates on several summatory functions involving the Moebius function

Abstract

We prove that | d≤x μ(d)/d| log x ≤ 1/69 when x ≥ 96 955 and deduce from that:for every x > q ≥ 1. We also give better constants when x/q is larger. Furthermore we prove that |1 − d≤x μ(d) log(x/d)/d| ≤ 3 14 / log x and several similar bounds, from which we also prove corresponding bounds when summing the same quantity, but with the additional condition (d, q) = 1. We prove similar results for d≤x μ(d) log 2 (x/d)/d, among which we mention the bound | d≤x μ(d) log 2 (x/d)/d − 2 log x + 2γ 0 | ≤ 5 24 / log x, where γ 0 is the Euler constant. We complete this collection by bounds such as d≤x,We also provide all these bounds with variations where 1/ log x is replaced by 1/(1 + log x).

Explicit average orders: news and problems

“…Rather than expanding on this subject, the author prefers to concentrate here on one application. Here is an identity proved in [37] by following [4] and [3]. For every x ě 1, we have…”

“…[37]). When D ě 463 421, we hav졡ˇÿdďD µpdq{dˇˇˇˇď 0.0144 log D´0.1 plog Dq 2 .When D ě 97 000, we haveˇˇˇˇÿ…”

Quantitative steps in the Axer–Landau equivalence theorem

“…Inferring a quantitative error term (here, simply a bound) for M (x) from the one of ψ(x) − x has received attention. Let us mention the work [7] of Kienast,[16] of Schoenfeld and [5] of El Marraki, although the last two authors do not present their investigation in this perspective; and lately the paper [13]. The answers are up to now rather unsatisfactory.…”

“…First, as noticed by Landau, inferring an error term for M (x) once we have one for m(x) is routine. Secondly, a path using identities has been investigated by Balazard [3] (see [2] for a French translation) and in [13].…”

“…Indeed, the latest best bound for 1/ζ(s) is due to Trudgian [17] and would put a constant at least of size 10 7 in front of x −1/(8 log log x) . When the emphasis is on this aspect, the path of identities taken in [3] and [13] remains a better choice.…”