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arXiv · 1804.09850

Explicit Bounds for $L$-Functions on the Edge of the Critical Strip

Allysa Lumley
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Abstract

Assuming GRH and the Ramanujan-Petersson conjecture we prove explicit bounds for $L(1,f)$ for a large class of $L$-functions $L(s,f)$, which includes $L$-functions attached to automorphic cuspidal forms on $GL(n)$. The proof generalizes work of Lamzouri, Li and Soundararajan. Furthermore, the main results improve the classical bounds of Littlewood \[(1+o(1))\left(\frac{12e^γ}{π^2}\log\log C(f)\right)^{-d} \leq |L(1,f)|\leq (1+o(1))\Big(2e^γ\log\log C(f)\Big)^d,\] where $C(f)$ is the analytic conductor of $L(s,f)$.

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Allysa Lumley. 2018-04-26. Explicit Bounds for $L$-Functions on the Edge of the Critical Strip. https://doi.org/10.1016/j.jnt.2018.01.001

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