## 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.

### MSC:

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

### Citations:

Zbl 0977.11033; Zbl 0744.11041; Zbl 0036.18603; Zbl 0977.11035
Full Text:

### References:

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