Explicit Estimates: From Λ(n) in Arithmetic Progressions to Λ(n)/n

ABSTRACT

We denote by the sum of Λ(n)/n for all n ≤ x and congruent to and similarly by ψ(x; q, a) the sum of Λ(n) over the same set. We show that the error term in 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 when x ≥ H. Previous methods had at best instead. We further compute the asymptotics for the L²-average of a quantity closely related to C(q, a).

KEYWORDS:

• prime numbers in arithmetic progressions
• explicit formulae
• explicit estimates

2000 AMS Subject Classification:

• Primary 11N05, 11M06, 11N13
• Secondary 11N37, 11A25