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

Abstract

General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms Explicit estimates: from Λ(n) in arithmetic progressions to Λ(n)/nMarch 5, 2015 AbstractWe denote byψ(x; q, a) the sum of Λ(n)/n for all n ≤ x and congruent to a mod q and similary by ψ(x; q, a) the sum of Λ(n) over the same set. We show that the error term inψ(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 primitive characters modulo q up to height H, this error term is bounded by O(e −H ) when x ≥ H. Previous methods had at best O(1/H) instead. We further compute asymptotics for the L 2 -average of quantity closely related to C(q, a).

Explicit average orders: news and problems

“…Recall that we are seeking quantitative results. In the case of the primes, it took the author quite a while, but D. Platt and the present author finally cleared the situation in [34] and [32]. Concerning the Moebius function, A. Axer in [1] produced a qualitative answer.…”

“…These computations concern also the zeros of L-functions, and this enables us to handle the distribution of primes in arithmetic progressions to some fixed modulus; the situation is less satisfactory there but getting tamed. Authors are now going towards derived quantities, e. g., ř logppq{p and Euler products, like O. Bordellès in [5], the present author in [32], the author's former student R. Mawia in [26] and P. Dusart in [16].…”

Quantitative steps in the Axer–Landau equivalence theorem

“…The Landau Theorem referred to above answers this question from a qualitative viewpoint. I addressed the question of a quantitative version of it in [12], and more completely together with D. Platt in [11], with applications to explicit estimates: the idea is to concentrate on ψ(x) − x and to automatically derive estimates forψ(x) − (log x − γ) (and for similar quantities with primes in arithmetic progressions). The results obtained are not (conjecturally) optimal but are still rather strong, both from a theoretical and numerical point of view; see below.…”

“…Though Theorem 1.1 of [11] is a consequence of the above, getting a statement numerically as good as Theorem 1.2 of the same paper is difficult, if at all possible, with the method we develop here.…”