Bounds for moments of Dirichlet \(L\)-functions to a fixed modulus (Q6582333)

Let \(\chi\) be a Dirichlet character modulo \(q\) with \(q\not\equiv2\pmod 4\). Denote \(\phi^\ast(q)\) be the number of primitive characters modulo \(q\). In this paper the author improves upper and lower bounds towards the widely believed asymptotic\N\[\N{\sum\limits_{\chi\pmod q}}^\ast\quad \vert L(\tfrac12,\chi)\vert^{2k}\sim C_k \phi^\ast(q)(\log q)^{k^2}\N\]\Nfor all \(k\ge0\), where \(\sum^\ast\) is the sum over primitive Dirichlet characters modulo \(q\) with \(C_k\) an explicit constant.\N\NIn the first theorem he obtains a lower bound\N\[\N{\sum\limits_{\chi\pmod q}}^\ast\quad \vert L(\tfrac12,\chi)\vert^{2k}\gg_k \phi^\ast(q)(\log q)^{k^2}\N\]\Nfor all real \(k\ge0\). This bound was first proved by \textit{Z. Rudnick} and \textit{K. Soundararajan} [Proc. Natl. Acad. Sci. USA 102, No. 19, 6837--6838 (2005; Zbl 1159.11317)] for all rational \(k\ge1\), and it was extended to all real \(k\ge1\) by \textit{M. Radziwiłł} and \textit{K. Soundararajan} [Mathematika 59, No. 1, 119--128 (2013; Zbl 1273.11128)] and to rational \(0<k<1\) by \textit{V. Chandee} and \textit{X. Li} [Int. Math. Res. Not. 2013, No. 19, 4349--4381 (2013; Zbl 1352.11067)].\N\NThe second theorem gives an upper bound\N\[\N{\sum\limits_{\chi\pmod q}}^\ast\quad \vert L(\tfrac12,\chi)\vert^{2k}\ll_k \phi^\ast(q)(\log q)^{k^2}\N\]\Nfor all real \(0\le k\le1\). Previously conditional results under GRH were known. Radzieill and Soundararajan created a method to obtain unconditional estimates, which was applied to the moments of quadratic twists of \(L\)-functions attached to elliptic curves. The article under review gives the first unconditional results for Dirichlet \(L\)-functions.