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Hybrid bounds for twists of $GL(3)$ $L$-functions

Qingfeng Sun

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Let $\pi$ be a Hecke-Maass cusp form for $SL(3,\mathbb{Z})$ and $\chi=\chi_1\chi_2$ a Dirichlet character with $\chi_i$ primitive modulo $M_i$. Suppose that $M_1$, $M_2$ are primes such that $\max\{\!(M|t|)^{\!1/3+2\delta/3\!},M^{2/5}|t|^{-9/20\!}, M^{1/2+2\delta}|t|^{-3/4+2\delta}\}(M|t|)^{\varepsilon\!}\!\lt\!M_1\!\lt\! \min\{ (M|t|)^{2/5\!},$ $(M|t|)^{1/2-8\delta}\}(M|t|)^{-\varepsilon}$ for any $\varepsilon\!>\!0$, where $M\!=\!M_1M_2$, $|t|\!\geq\! 1$, and $0\lt\delta\lt 1/52$. Then we have $$ L\left(\frac{1}{2}+it,\pi\otimes \chi\right)\ll_{\pi,\varepsilon} (M|t|)^{3/4-\delta+\varepsilon}. $$

Article information

Publ. Mat., Volume 64, Number 1 (2020), 75-102.

Received: 17 January 2018
First available in Project Euclid: 3 January 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

hybrid bounds $GL(3)$ $L$-functions twists


Sun, Qingfeng. Hybrid bounds for twists of $GL(3)$ $L$-functions. Publ. Mat. 64 (2020), no. 1, 75--102. doi:10.5565/PUBLMAT6412003.

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