2014/12/01 by Olivier Ramaré · 1 citation
Mathematics · #Analytic Number Theory Research #Mathematical Approximation and Integration #Analytic and geometric function theory
paper · pdf · doi:10.1090/s0025-5718-2014-02914-1
We prove that |∑ d≤ xμ (d)/d|log x≤ 1/69 when x≥ 96 955 and deduce from that: |\textstyle ∑ _ \\substack d≤ x,
(d,q)=1 .μ (d)/d |log (x/q)≤ \tfrac 45 q/φ (q) for every x>q≥ 1. We also give better constants when x/q is larger. Furthermore we prove that |1-∑ d≤ xμ (d)log (x/d)/d|≤ \tfrac 314/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≤ xμ (d)log 2(x/d)/d, among which we mention the bound |∑ d≤ xμ (d)log 2(x/d)/d-2log x+2γ 0|≤ \tfrac 524/log x, where γ 0 is the Euler constant. We complete this collection by bounds such as \textstyle |∑ _ \\substack d≤ x,
(d,q)=1 .μ (d) |/x≤ \tfrac qφ (q)/log (x/q). We also provide all these bounds with variations where 1/log x is replaced by 1/(1+log x).