Asymptotic expansions of the mean values of Dirichlet \(L\)-functions. III. (English) Zbl 0977.11034

For Part II, cf. the preceding review Zbl 0977.11033.
Let \(\chi\) be a Dirichlet character to modulus \(q\), and \(L(s,\chi)\) the corresponding Dirichlet \(L\)-function. K. Matsumoto and the author [Math. Z. 208, 23-39 (1991; Zbl 0744.11041)] showed that the mean square \[ \sum_{\chi\pmod q}|L(s,\chi)|^2 \] has an asymptotic expansion in the descending order of \(q\) for \(0< \operatorname {Re}s< 1\). In the paper under review, the mean square of derivatives of \(L\)-functions \[ \sum_{\chi\pmod q}|L^{(h)} (s,\chi)|^2 \quad (h= 0,1,2,\dots) \] are investigated by refining the argument of the paper mentioned above. As a corollary of the main formula, asymptotic expansions for the special cases \(\operatorname {Re}s= 1/2\) and \(s=1\) with \(h=1\) are explicitly given. An essential tool to deduce error estimates for the remainder term is a suitably modified lemma of F. V. Atkinson [Acta Math. 81, 353-376 (1949; Zbl 0036.18603), Lemma 2.3] which asymptotically evaluates certain exponential integrals.


11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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