Abstract
It is proved that as T → ∞, uniformly for all positive integers ℓ ⩽ (log3T)/(log4T), we have maxT ⩽ t ⩽ 2T(ζ(ℓ),(1, + ,i,t)) ⩾ (Yℓ, + ,o,(1))(log2,T)ℓ+1 , where Yℓ = ∫0∞uℓρ(u)du. Here, ρ(u) is the Dickman function. We have Yℓ > eγ/(ℓ + 1) and log Yℓ = (1 + o(1))ℓlogℓ when ℓ → ∞, which significantly improves previous results in [17, 40]. Similar results are established for Dirichlet L-functions. On the other hand, when assuming the Riemann hypothesis and the generalized Riemann hypothesis, we establish upper bounds for |ζ(ℓ)(1 + it)| and |L(ℓ)(1, χ)|. Furthermore, when assuming the Granville-Soundararajan conjecture is true, we establish the following asymptotic formulas: [Formula: see text] where q is prime and ℓ ∈ N is given.