A conditional bound for the least prime in an arithmetic progression

TL;DR

Under the assumption of the generalized Lindelöf hypothesis, the paper derives a nearly optimal upper bound on the least prime in an arithmetic progression, improving understanding of prime distribution under certain hypotheses.

Contribution

The paper establishes a conditional bound on the least prime in an arithmetic progression assuming the generalized Lindelöf hypothesis, approaching the classical conjectured bounds.

Findings

Bound p ≪ q^{2+ε} for the least prime in an arithmetic progression

Nearly matches the classical estimate under the generalized Riemann hypothesis

Provides a conditional result based on a widely believed hypothesis

Abstract

Assuming the generalized Lindel\"of hypothesis for Dirichlet $L$-functions, we establish that the least prime $p\equiv a\pmod{q}$ satisfies $p\ll_{\varepsilon} q^{2+\varepsilon}$. This achieves a bound that nearly matches the classical estimate implied by the generalized Riemann hypothesis.