Explicit Estimates: From Λ(n) in Arithmetic Progressions to Λ(n)/n. Issue 1 (2nd January 2017)

• Record Type: Journal Article
• Authors: Platt, D. J.; Ramaré, O.
• Is Part Of: Experimental mathematics. Volume 26:Issue 1(2017)
• Page Start: 77
• Page End: 92
• Publication Date: 2017-01-02
• Subjects: prime numbers in arithmetic progressions -- explicit formulae -- explicit estimates; Primary 11N05, 11M06, 11N13 -- Secondary 11N37, 11A25; Mathematics -- Periodicals; Mathematics -- Research -- Periodicals; 510.724
• DOI: 10.1080/10586458.2015.1123124
• Languages: English
• ISSNs: 1058-6458

Abstract

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 L 2 -average of a quantity closely related to C ( q, a ).