TY - JOUR
T1 - On large values of L(σ,χ)
AU - Aistleitner, Christoph
AU - Mahatab, Kamalakshya
AU - Munsch, Marc Alexandre
AU - Peyrot, Alexandre
PY - 2018/12
Y1 - 2018/12
N2 - 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.
AB - 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.
U2 - 10.1093/qmath/hay067
DO - 10.1093/qmath/hay067
M3 - Article
SN - 0033-5606
VL - 70
SP - 831
EP - 848
JO - The Quarterly Journal of Mathematics
JF - The Quarterly Journal of Mathematics
IS - 3
ER -