Explicit estimates on the summatory functions of the Moebius function with coprimality restrictions
نویسنده
چکیده
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.
منابع مشابه
Explicit estimates on several summatory functions involving the Moebius function
We prove that | ∑ d≤x μ(d)/d| log x ≤ 1/69 when x ≥ 96 955 and deduce from that: ∣ ∣ ∣ ∣ ∑{ d≤x, (d,q)=1 μ(d)/d ∣ ∣ ∣ ∣ log(x/q) ≤ 4 5 q/φ(q) 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 ...
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