Gaps of Smallest Possible Order between Primes in an Arithmetic Progression

Abstract

Let $t\in \mathbb{N}$, $\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q\le x^{5/12-\eta }$, $q$ not a multiple of the conductor of the exceptional character ${\chi }^{\ast }$ (if it exists). Suppose further that, \[ \max \{p : p | q \} < \exp (\frac{\log x}{C \log \log x}) \; \; {and} \; \; \prod_{p | q} p < x^δ, \] where $C$ and $\delta$ are suitable positive constants depending on $t$ and $\eta$. Let $a\in \mathbb{Z}$, $(a,q)=1$ and \[ \mathcal{A} = \{n \in (x/2, x]: n \equiv a \pmod{q} \} . \] We prove that there are primes $p_1<p_2<...<p_t$ in $\mathcal{A}$ with \[ p_t - p_1 \ll qt \exp (\frac{40 t}{9-20 θ}) . \] Here $\theta =(\log q)/\log x$.

18 pages

References in corpus (1)

• Gaps between Primes in Beatty Sequences