On large values of L(σ,χ)

Christoph Aistleitner*, Kamalakshya Mahatab, Marc Alexandre Munsch, Alexandre Peyrot

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In recent years, a variant of the resonance method was developed which allowed to obtain improved Ω-results for the Riemann zeta function along vertical lines in the critical strip. In the present paper, we show how this method can be adapted to prove the existence of large values of |L(σ,χ)| in the range σ∈(1/2,1]⁠, and to estimate the proportion of characters for which |L(σ,χ)| is of such a large order. More precisely, for every fixed σ∈(1/2,1)⁠, we show that for all sufficiently large q⁠, there is a non-principal character χ(modq) such that log∣∣L(σ,χ)∣∣≥C(σ)(logq)1−σ(loglogq)−σ⁠. In the case σ=1⁠, we show that there is a non-principal character χ(modq) for which |L(1,χ)|≥eγ(log2q+log3q−C)⁠. In both cases, our results essentially match the prediction for the actual order of such extreme values, based on probabilistic models.
Original languageEnglish
Pages (from-to)831-848
JournalThe Quarterly Journal of Mathematics
Issue number3
Publication statusPublished - Dec 2018


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