A Note on the Linnik's constant

TL;DR

This paper improves bounds on the least prime in arithmetic progressions with moduli having bounded cubic parts by combining Heath-Brown's method and Burgess's bounds for L-functions.

Contribution

It introduces a new bound on the least prime in arithmetic progressions for certain moduli, enhancing previous results by integrating advanced analytic techniques.

Findings

Established that P(a,q) is bounded by q^{4.5} for specific q and a.

Combined Heath-Brown's method with Burgess's bounds for L-functions.

Provides a new explicit bound on the least prime in arithmetic progressions.

Abstract

Let $P(a,q)$ be the least prime in the arithmetic progression $\{n\equiv a(mod\ q)\}$. In this note, when $q$ has bounded cubic part and $(a,q)=1$, we combine the Heath-Brown's method and the Burgess's bounds for L-functions to obtain $ P(a,q)\ll q^{4.5}.$