Explicit estimates: from Λ(n) in arithmetic progressions to Λ(n)/n

We denote by ψ(x; q, a) the sum of Λ(n)/n for all n≤x and congruent to a mod q and similarly by ψ(x; q, a) the sum of Λ(n) over the same set. We show that the error term in ψ(x; q, a) − (log x)/ϕ(q) − C(q, a) for a suitable constant C(q, a) can be controlled by that of ψ(y; q, a)−y/ϕ(q) for y of size x, up to a small error term. As a consequence, if a partial generalized Riemann hypothesis has been verified for the L-functions attached to characters modulo q up to height H, this error term is bounded by O(e^−H/8) when x ≥ H. Previous methods had at best O(1/H) instead. We further compute the asymptotics for the L²-average of a quantity closely related to C(q, a).