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

Abstract

Abstract Assuming the generalized Riemann hypothesis, we provide explicit upper bounds for moduli of $$\log {\mathcal {L}(s)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>log</mml:mo> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and $$\mathcal {L}'(s)/\mathcal {L}(s)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> in the neighbourhood of the 1-line when $$\mathcal {L}(s)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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.

Locations

• Mediterranean Journal of Mathematics - View - PDF