Extremal values of Dirichlet L - functions in the half - plane of absolute convergence par

We prove that for any real θ there are infinitely many values of s = σ + it with σ → 1+ and t→ +∞ such that Re {exp(iθ) logL(s, χ)} ≥ log log log log t log log log log t +O(1). The proof relies on an effective version of Kronecker’s approximation theorem. 1. Extremal values Extremal values of the Riemann zeta-function in the half-plane of absolute convergence were first studied by H. Bohr and Landau [1]. Their results rely essentially on the diophantine approximation theorems of Dirichlet and Kronecker. Whereas everything easily extends to Dirichlet series with real coefficients of one sign (see [7], §9.32) the question of general Dirichlet series is more delicate. In this paper we shall establish quantitative results for Dirichlet L-functions. Let q be a positive integer and let χ be a Dirichlet character mod q. As usual, denote by s = σ + it with σ, t ∈ R, i2 = −1, a complex variable. Then the Dirichlet L-function associated to the character χ is given by