The generalized Lindelöf hypothesis for Dirichlet $L$ -functions

Let $\chi\pmod{q}$ be a Dirichlet character, and write $s=\sigma+it$ . Generalized Lindelöf hypothesis. For every $\varepsilon>0$ , if $\sigma\geq 1/2$ and $|s-1|\geq 1/2$ , then

L(\sigma+it,\chi)\ll_{\varepsilon}(q(1+|t|))^{\varepsilon}.

This is a standard conjectural bound for Dirichlet $L$ -functions, extending the critical-line formulation to the half-plane $\sigma\geq 1/2$ away from the pole at $s=1$ . The paper assumes this hypothesis to obtain a near-quadratic conditional bound for the least prime in an arithmetic progression; it remains open.

Sources & referencesView supporting material

Primary source

Matías Bruna, “A conditional bound for the least prime in an arithmetic progression”, arXiv:2603.25612 (2026).

Additional references

6 papers in this index state this conjecture (2024–2026). The statement above is taken from the most recent of them; the others are arXiv:2509.01602, arXiv:2505.02968, arXiv:2405.14834, arXiv:2402.03475, arXiv:2401.05052.

analytic-number-theory dirichlet-l-functions prime-number-theorem

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Progress summary

Refreshed 2 days ago

Open

The generalized Lindelöf hypothesis remains unproved: recent work either assumes it or explicitly says its formal calculation is not a proof.

The conjecture gives a nearly minimal uniform growth bound for Dirichlet $L$ -functions in the half-plane $4\sigma\geq 1/2\u00024$ . It remains open; no verified proof or counterexample was found.

Recent developments in 2026

A March 2026 paper by Matías Bruna proves a near-quadratic bound for the least prime in an arithmetic progression, but only conditional on the generalized Lindelöf hypothesis; it does not advance the conjecture itself.
The preprint An approach to the Lindelöf Hypothesis for Dirichlet $L$ -functions presents a formal calculation suggesting the critical-line estimate, but explicitly says convergence and rearrangement of sums are unjustified and derives an incorrect identity; it is not a proof.

Current status (as of August 2026): The generalized Lindelöf hypothesis for Dirichlet $L$ -functions remains open, with no verified proof, counterexample, or substantive progress recorded in the retrieved sources.

Sources

No solutions have been posted yet.