Gaps of Smallest Possible Order between Primes in an Arithmetic Progression

TL;DR

This paper establishes bounds on the gaps between primes within an arithmetic progression under certain conditions on the modulus and the distribution of primes, extending previous results in analytic number theory.

Contribution

It provides new bounds on prime gaps in arithmetic progressions for large moduli with specific restrictions, advancing understanding of prime distribution in such sequences.

Findings

Existence of primes in arithmetic progressions with bounded gaps

Bounds depend on the modulus and the number of primes t

Results hold for large x under specified conditions

Abstract

Let $t \in \mathbb{N}$, $\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-\eta}$, $q$ not a multiple of the conductor of the exceptional character $\chi^*$ (if it exists). Suppose further that, \[ \max \{p : p | q \} < \exp (\frac{\log x}{C \log \log x}) \; \; {and} \; \; \prod_{p | q} p < x^{\delta}, \] 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 \[ p_t - p_1 \ll qt \exp (\frac{40 t}{9-20 \theta}) . \] Here $\theta = (\log q) / \log x$.