A note on the mean value of the zeta and L-functions. II
In the present note [for Part I see ibid. 61, 222-224 (1985; Zbl 0573.10027)] the author considers, for \(\chi\) a primitive character mod q, \[ E(T,\chi)=\int^{T}_{0}| L(+it,\chi)|^ 2 dt-(\phi (q)/q)T\{\log (qT/2\pi)+2\gamma +2\sum_{p| q}(\log p)/(p-1)\}. \] For \(q=1\) (i.e., the case of the Riemann zeta-function \(\zeta\) (s)) the above function has been explicitly determined by \textit{F. V. Atkinson} [Acta Math. 81, 353-376 (1949; Zbl 0036.186)]. The main aim of the present work is not to determine an explicit expression for E(T,\(\chi)\), but finding upper bound estimates of E(T,\(\chi)\) which are uniform in q and T. The author only sketches the proof of the remarkable result that if \(\chi\) is a non-principal character mod q and q is a prime, then uniformly \[ (1)\quad E(T,\chi)\ll ((qT)^{1/3}+q^{1/2})(\log qT)^ 4. \] The proof of (1) is based on five lemmas, some of which are of independent interest. The case when q is composite is promised to be treated subsequently. The bound in (1) is very good, and it is the sharpest one hitherto. It improves the corresponding bound which may be found in the recent work of \textit{T. Meurman} [A generalization of Atkinson's formula to L-functions, Acta Arith. (to appear) (Zbl 0561.10019)] whose main aim was obtaining an explicit formula for E(T,\(\chi)\) in the spirit of Atkinson, and whose methods of proof differ from those used by the present author.
scientific article; zbMATH DE number 3933165