Estimates for L-functions in the Critical Strip Under GRH with Effective Applications

Abstract

Assuming the generalized Riemann hypothesis, we provide explicit upper bounds for moduli of log {mathcal {L}(s)} and mathcal {L}'(s)/mathcal {L}(s) in the neighbourhood of the 1-line when mathcal {L}(s) are the Riemann, Dirichlet and Dedekind zeta-functions. To do this, we generalize Littlewood’s well-known conditional result to functions in the Selberg class with a polynomial Euler product, for which we also establish a suitable convexity estimate. As an application, we provide conditional and effective estimates for the Mertens function.