On the argument of L-functions

Abstract

For L(·, π) in a large class of L-functions, assuming the generalized Riemann hypothesis (GRH), we show an explicit bound for the function ${S_1}\left( {t,\pi } \right) = \frac{1}{\pi }\int_{1/2}^\infty {\log \left| {L\left( {\sigma + it,\pi } \right)} \right|} d\sigma$, expressed in terms of its analytic conductor. This enables us to give an alternative proof of the most recent (conditional) bound for ${S_1}\left( {t,\pi } \right) = \frac{1}{\pi }$ arg $L\left( {\frac{1}{2} + it,\pi } \right)$, which is the derivative of S ₁(·, π) at t. Additional applications include bounds for the maximal multiplicity of a zero of an L-function and bounds for the maximum gap between consecutive zeros of an L-function, both under GRH.