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 L(s) and L ′ (s)/L(s) in the neighbourhood of the 1-line when L(s) are the Riemann, Dirichlet and Dedekind zetafunctions. 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 estimate for the Mertens function.

A modular relation involving non-trivial zeros of the Dedekind zeta function, and the Generalized Riemann Hypothesis

“…Indeed, the conditions (5.1), (5.2) and the functional equation (equations (5.3) to (5.5)) in [11, p. 94] essentially reiterate that ζ K (s) lies in the Selberg class of functions with polynomial Euler product. This is well-known, see for instance, p.5 of [19]. The condition (3) in [11, p. 94] also holds since ζ K (s) has conductor |d K |, as mentioned on p. 125 of [11].…”

Conditional estimates for the logarithmic derivative of Dirichlet $L$-functions

“…Recently, explicit and conditional results on the logarithmic derivative of L-functions in the Selberg class of functions with a polynomial Euler product were announced in [18]. However, they are worse than (1.1).…”