We prove that $|{\sum }_{d\le x,(d,q)=1}\mu \left(d\right)/d|\le 2.4(q/\phi \left(q\right))/log(x/q)$ for every x > q ≥ 1, and similar estimates for the Liouville function. We also give better constants when x/q is large.,

Let χ be a primitive Dirichlet character of conductor q and denote by L(z,χ) the associated L-series. We provide an explicit upper bound for |L(1,χ)| when 3 divides q.