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The mean square of Dirichlet L-functions. (A generalization of Balasubramanian’s method). (English) Zbl 0724.11039
Let $$\chi$$ be a primitive Dirichlet character mod q, and L(s,$$\chi$$) the corresponding Dirichlet L-function. Let $E(T,\chi)=\int^{T}_{0}| L(1/2+it,\chi)|^ 2 dt-\frac{\phi (q)}{q}(\log \frac{qT}{2\pi}+2\gamma -1+2\sum_{p| q}\frac{\log p}{p-1})T$ denote the error term in the mean square formula for $$L(1/2+it,\chi)$$, where $$\gamma$$ is Euler’s constant and $$\phi$$ is the Euler function. Y. Motohashi proved [Proc. Japan Acad., Ser. A 61, 313-316 (1985; Zbl 0583.10024)], among other things, that $E(T,\chi)\ll ((qT)^{1/3}+q^{1/2})(\log qT)^ 4$ uniformly for $$T\geq 2$$ and any q. His argument was a natural extension of the work of F. V. Atkinson [Acta Math. 81, 353-376 (1949; Zbl 0036.186)] on the mean square of the Riemann zeta-function. Prior to Motohashi’s work, the author [Proc. Japan Acad., Ser. A 58, 443-446 (1982; Zbl 0514.10030) and ibid. 65, 344 (1989; Zbl 0699.10059)], investigated the same problem by a different method. His basic idea was to combine the method of R. Balasubramanian [Proc. Lond. Math. Soc., III. Ser. 36, 540-576 (1978; Zbl 0375.10025)] with Weil’s estimate for Kloosterman sums. Unfortunately his argument contained a serious error, which led to an erroneous result.
In the article under review the author sketches the corrected argument, which gives $E(T,\chi)\ll (qT)^{1/3+\epsilon}$ when q is odd and $$T\gg q^{20}$$. This is the right generalization of Balasubramanian’s theorem, although it is still weaker than Motohashi’s result. A lengthy manuscript exists, in which all the details of the proof are worked out.
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$