Une région explicite sans zéros pour la fonction ζ de Riemann

“…As a consequence, we find that |b(χ)| ≤ 14 + 54 log q + 14 log 2 q (χ not exceptional.) (28) This bound can be improved in several ways, for instance on invoking the improved zero-free region for L-function proved in [13] (see also [14]). This result is heavily influenced by the small zeros for which a better result may be known.…”

Section: Acknowledgementmentioning

confidence: 99%

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

Platt

Ramaré

2016

Experimental Mathematics

View full text Add to dashboard Cite

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).

show abstract

“…Let π(x) be the number of primes up to x. Hadamard and de la Vallée Poussin proved the prime number theorem which states that π(x) is asymptotic to Li(x) = See [18] and Section 1 of [7]. The constant R 0 has been improved by many authors, particularly by Rosser [13,14,15], Schoenfeld [14,15], Stechkin [16], Ford [1,2], and Kadiri [7]. Recently Kadiri improved significantly on R 0 .…”

Section: Introductionmentioning

confidence: 99%