The mean twelfth power of Dirichlet L-functions on the critical line. (English) Zbl 0538.10034

D. R. Heath-Brown [Q. J. Math., Oxf. II. Ser. 29, 443-462 (1978; Zbl 0394.10020)] has shown that for \(T\geq 2\) \[ (1)\quad \int^{T}_{0}| \zeta(1/2+it)|^{12}\quad dt\quad<<\quad T^ 2 \log^{17}T. \] His proof depends on Atkinson’s formula for \(\int^{T}_{0}| \zeta(1/2+it)|^ 2\) dt. M. Jutila [J. Number Theory 18, 135-156 (1984; Zbl 0533.10034)] has established a transformation formula for \(\sum_{M_ 1\leq n\leq M_ 2}d(n)n^{-1/2- it},\) where d(n) is the divisor function, \(M_ 1\), \(M_ 2\) are near t/2\(\pi\) and \(M_ 1<t/2\pi<M_ 2\). As he pointed out, this can be used instead of Atkinson’s formula in proving (1).
In the present paper Jutila’s formula is generalized to sums \(\sum_{M_ 1\leq n\leq M_ 2}\chi(n)d(n)n^{-1/2-it},\) where \(\chi\) (n) is a Dirichlet character mod q, \(M_ 1\), \(M_ 2\) are near qt/2\(\pi\) and \(M_ 1<qt/2\pi<M_ 2\). The result is applied to obtain a large values theorem for L-functions. This implies an extension of (1), viz. \[ \sum_{\chi \quad mod q}\int^{T}_{-T}| L(1/2+it,\chi)|^{12}\quad dt\quad<<_{\epsilon}\quad q^ 3 T^{2+\epsilon}. \] The above results are applied to deduce estimates for the zero counting function \(\sum_{\chi \quad mod q}N(\alpha,T,\chi),\) which generalize previous estimates of D. R. Heath-Brown [J. Lond. Math. Soc., II. Ser. 19, 221-232 (1979; Zbl 0393.10043)] and are new if q is sufficiently small compared with T.


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