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

Abstract

We prove that | d≤x, (d,q)=1 µ(d)/d| ≤ 2.4 (q/ϕ(q))/ log(x/q) for every x > q ≥ 1 and similar estimates for the Liouville functions. We give also better constants when x/q is larger.

Explicit estimates on several summatory functions involving the Moebius function

“…This improves on the corresponding estimates proved in [19]. Prior to this paper, the sole estimate on m q (x) seems to be [11, Lemma 10.2] which bounds |m q (x)| uniformly by 1.…”

“…We proposed in [19] an approach via the Liouville function λ(n) (the completely multiplicative function that is 1 on integers that have an even number of prime factors, counted with multiplicity, and −1 otherwise). Such an approach splits the evaluation into three steps: expressing m q (x) in terms of q , where…”

Explicit average orders: news and problems

“…It still seems to decrease slowly. We adapted the GP/Pari script described in the proof of Lemma 2.1 of [36] and let it run for some days (on a desktop computer having only 8 Gigabytes of RAM, the computation was split to intervals of length 2¨10 7 ).…”

On a logarithmic sum related to the Selberg sieve

“…The error term magnitude in the theorem above has been obtained by relying on estimations by Balazard [1], Bordèlles [3], El-Marraki [6], Helfgott [8, §6] and Ramaré [15], [16], among others; nonetheless, when using non-explicit tools, one should expect an error of magnitude e −w √ log(U ) for some constant w > 0. Furthermore, in order to derive Theorem 1, one must abandon the classical prime-number-theorem-like approach (as in [7], for example), as it gives ineffective estimations, or explicit that involve huge numbers, and thus are inconvenient or impractical.…”