A conditional bound for the least prime in an arithmetic progression

Matías Bruna

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arXiv · 2603.25612

A conditional bound for the least prime in an arithmetic progression

Matías Bruna

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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.

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Matías Bruna. 2026-03-26. A conditional bound for the least prime in an arithmetic progression. https://arxiv.org/abs/2603.25612

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