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(2) |ζ(σ + it)| ≤ c1t100(1−σ) 3/2 log t for 1/2 ≤ σ ≤ 1, t ≥ 2 and a positive constant c1. More generally, one can deduce from (1) that if L(s, χ) denotes the Dirichlet L-function associated with the Dirichlet character χ (mod k), k ≥ 1, then (3) |L(σ + it, χ)| ≤ c2k1−σt100(1−σ) 3/2 log t+ k1−σ log k for the same range as (2) and a positive constant c2. The bounds of the type (2) or (3) have existed for a long time in the literature and have various applications (zero-free regions, a problem of Dirichlet divisors in number fields, the order of the Dedekind zeta function of a quadratic field, and so on).

Refinement of an estimate for the Hurwitz zeta function in a neighbourhood of the line σ = 1