A note on the mean value of the zeta and L-functions. III
This is Part III of a series of notes with the same title [Part I, ibid. 61, 222-224 (1985; Zbl 0573.10027); Part II, ibid. 61, 313-316 (1985; Zbl 0583.10024)]. The author now generalizes the twelfth power moment for \(| \zeta (1/2+it)|\) of \textit{D. R. Heath-Brown} [Q. J. Math., Oxf. II. Ser. 29, 443-462 (1978; Zbl 0394.10020)] to \(| L(1/2+it,\chi |\). The main result is the bound \[ \int^{T}_{0}| L(1/2+it,\chi)|^{12} dt\quad \ll \quad ((qT)^ 2+q^ 3)(\log qT)^{36},\quad (T\geq 1) \] proved under the assumption that \(\chi\) is a non-principal character mod q, q a prime (the last restriction being of a technical nature). The result may be compared with that of \textit{T. Meurman} [Ann. Acad. Sci. Fenn., Ser. A I, Diss. 52, 44 p. (1984; Zbl 0538.10034)], who proved by different methods that, for any \(q\geq 1\) and \(T\geq 2\), \[ \sum_{\chi mod q}\int^{T}_{-T}| L(1/2+it,\chi)|^{12} dt\quad \ll_{\epsilon}\quad q^ 3 T^{2+\epsilon}. \]
scientific article; zbMATH DE number 3950637