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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.
Reviewer: A.Ivić (Beograd)

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
Full Text: DOI
[1] Atkinson, F. V.: The mean-value of the Riemann zeta function. Acta Math., 81, 353-376 (1949). · Zbl 0036.18603 · doi:10.1007/BF02395027
[2] Balasubramanian, R.: An improvement on a theorem of Titchmarsh on the mean square of |C(l/2+i*)|. Proc. London Math. Soc, (3) 36, 540-576 (1978). · Zbl 0375.10025 · doi:10.1112/plms/s3-36.3.540
[3] Kober, H.: Eine Mittelwertformel der Riemannschen Zetafunktion. Compositio Math., 3, 174-189 (1936). · Zbl 0014.25804 · numdam:CM_1936__3__174_0 · eudml:88619
[4] Matsumoto, K.: The mean square of Dirichlet L-functions. Proc. Japan Acad., 58A, 443-446 (1982). · Zbl 0514.10030 · doi:10.3792/pjaa.58.443
[5] Matsumoto, K.: On the mean square of Dirichlet L-functions. Surikaiseki Kenkyusho Kokyuroku, 496, Res. Inst. Math. Sci. Kyoto Univ., 123-148 (1983) (in Japanese). · Zbl 0514.10030 · doi:10.3792/pjaa.58.443
[6] Matsumoto, K.: Corrections to the mean square of Dirichlet L-functions. Proc. Japan Acad., 65A, 344 (1989). · Zbl 0699.10059 · doi:10.3792/pjaa.65.344
[7] Matsumoto, K.: xhe mean square of Dirichlet L-functions (A generalization of Balasubramanian’s method). Manuscript, 288 pp. (1990). · Zbl 0724.11039
[8] Meurman, T.: A generalization of Atkinson’s formula to L-functions. Acta Arith., 47, 351-370 (1986). · Zbl 0561.10019 · eudml:206036
[9] Motohashi, Y.: A note on the mean value of the zeta and L-functions. II. Proc. Japan Acad., 61 A, 313-316 (1985). · Zbl 0583.10024 · doi:10.3792/pjaa.61.313
[10] Motohashi, Y.: On the mean square of L-functions. Manuscript, 42pp.+llpp. (1986).
[11] Titchmarsh, E. C.: On van der Corput’s method and the zeta-function of Rie-mann (V). Quart. J. Math., Oxford, 5, 195-210 (1934). · Zbl 0010.01002
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