A note on the zeros of zeta and $L$-functions

Abstract

Let $\pi S(t)$ denote the argument of the Riemann zeta-function at the point $s=\frac{1}{2}+it$. 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.

References in corpus (2)

• Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function
• On the argument of $L$-functions

Cited by in corpus (7)

• Extremal functions in de Branges and Euclidean spaces
• On the argument of $L$-functions
• Bounding $S_n(t)$ on the Riemann hypothesis
• Extremal functions in de Branges and Euclidean spaces II
• Conditional estimates for the logarithmic derivative of Dirichlet $L$-functions
• Moments in the Chebotarev density theorem: general class functions
• On the lowest zero of Dedekind zeta function