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arXiv · 1503.00955

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

Emanuel Carneiro·Vorrapan Chandee·Micah B. Milinovich
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Abstract

Let $\pi S(t)$ denote the argument of the Riemann zeta-function at the point $s=\tfrac12+it$. 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.

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Emanuel Carneiro, Vorrapan Chandee, Micah B. Milinovich. 2015-03-03. A note on the zeros of zeta and $L$-functions. https://doi.org/10.1007/s00209-015-1485-9

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