On the Brun-Titchmarsh theorem

• Autorzy: James Maynard
• Języki publikacji: EN

Abstrakty

EN

The Brun-Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq ≥ 8 and q is sufficiently large. Moreover, we also produce a lower bound of size $x/(q^{1/2}ϕ(q))$ when logx/logq ≥ 8 and is bounded. Both of these bounds are essentially best-possible without any improvement on the Siegel zero problem.

Kategorie tematyczne

• 11N05: Distribution of primes
• 11M06: ζ ( s ) and L ( s , χ )
• 11M20: Real zeros of L ( s , χ ) ; results on L ( 1 , χ )
• 11N13: Primes in progressions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 157
• Numer: 3
• Strony: 249-296

Daty

wydano

2013

Twórcy

autor

James Maynard

• Mathematical Institute, 24-29 St Giles', Oxford, OX1 3LB, England

Identyfikatory

DOI

10.4064/aa157-3-3