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 $$\log {\mathcal {L}(s)}$$ log L ( s ) and $$\mathcal {L}'(s)/\mathcal {L}(s)$$ ′ / in neighbourhood 1-line when are Riemann, Dirichlet Dedekind zeta-functions. To do this, generalize Littlewood’s well-known conditional result to functions Selberg class with a polynomial Euler product, which also establish suitable convexity estimate. As an application, effective estimates Mertens function.