Petrow, Ian; Young, Matthew P. The fourth moment of Dirichlet \(L\)-functions along a coset and the Weyl bound. (English) Zbl 1544.11068 Duke Math. J. 172, No. 10, 1879-1960 (2023). Let \(\chi\) be a primitive Dirichlet character modulo \(q\). In this paper under review, the authors continue their previous work [Ann. Math. (2) 192, No. 2, 437–486 (2020; Zbl 1460.11111)] on the Weyl bound for Dirichlet \(L\)-functions of cube-free conductor. More precisely, they remove the cube-free hypothesis and establish the following Weyl bound without any restrictions on the conductor of \(\chi\) and for any \(\epsilon>0\) \[L\left(1/2+it,\chi\right)\ \ll_\varepsilon\ (q(1+|t|))^{\frac16+\varepsilon}\ q^{\frac16+\varepsilon}.\] The proof of the above estimate is derived from a Lindelöf-on-average upper bound for the fourth moment of Dirichlet \(L\)-functions of conductor \(q\) along a coset of the subgroup of characters modulo \(d\) when \(q^\ast \mid d\), where \(q^\ast\) is the least positive integer such that \(q^2 \mid ( q^\ast )^3\). Reviewer: Sami Omar (Sukhair) Cited in 9 Documents MSC: 11M06 \(\zeta (s)\) and \(L(s, \chi)\) 11F11 Holomorphic modular forms of integral weight 11F12 Automorphic forms, one variable 11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations Keywords:character sums; Dirichlet \(L\)-functions; moments of \(L\)-functions; subconvexity Citations:Zbl 1460.11111 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Link Link References: [1] N. ANDERSEN and E. M. \(Kı\) RAL, Level reciprocity in the twisted second moment of Rankin-Selberg L-functions, Mathematika 64 (2018), no. 3, 770-784. · Zbl 1448.11096 · doi:10.1112/s0025579318000256 [2] A. O. L. ATKIN and J. LEHNER, Hecke operators on \( \operatorname{\Gamma}_0(m)\), Math. Ann. 185 (1970), 134-160. · doi:10.1007/BF01359701 [3] A. O. L. ATKIN and W.-C. W. LI, Twists of newforms and pseudo-eigenvalues of W-operators, Invent. Math. 48 (1978), no. 3, 221-243. · doi:10.1007/BF01390245 [4] V. BLOMER, P. 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