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• # Artykuł - szczegóły

## Acta Arithmetica

2013 | 157 | 3 | 249-296

## On the Brun-Titchmarsh theorem

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.

249-296

wydano
2013

### Twórcy

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