A note on the zeros of zeta and L-functions

Let πS(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi S(t)$$\end{document} denote the argument of the Riemann zeta-function at the point s=12+it\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=\tfrac{1}{2}\,+\,it$$\end{document}. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for S(t). We discuss a generalization of this bound for a large class of L-functions including those which arise from cuspidal automorphic representations of GL(m) over a number field. We also prove a number of related results including bounding the order of vanishing of an L-function at the central point and bounding the height of the lowest zero of an L-function.