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Asymptotic expansions of the mean values of Dirichlet \(L\)-functions. (English) Zbl 0744.11041

Let \(L(s,\chi)\) be the Dirichlet \(L\)-function corresponding to a Dirichlet character \(\chi\bmod q\) (\(\geq 2\)) with a complex variable \(s=\sigma+it\). By \(\zeta(s)\), \(\phi(n)\) and \(\mu(n)\) we denote the Riemann zeta, the Euler and the Möbius function, respectively. In this paper we prove the following formulae for \(0<\sigma < 1\) and an arbitrary real \(t\): \[ \begin{split} \varphi(q)^{-1}\sum_{\chi(\bmod q)}| L(\sigma + it,\chi)|^ 2 =\zeta(2\sigma)\prod_{p\mid q}(1-p^{- 2\sigma})+ \\ +2q^{-2\sigma}\varphi(q)\Gamma(2\sigma-1)\zeta(2\sigma- 1)\hbox{ Re}\left\{{\Gamma(1-\sigma+it)\over \Gamma(\sigma+it)}\right\} +2q^{-2\sigma}\sum_{k\mid q}\mu(q/k) T(\sigma+it;k)\end{split} \] for \(\sigma\neq 1/2\), and \[ \begin{split} \varphi(q)^{-1}\sum_{\chi(\bmod q)}| L({1\over 2}+it;\chi)|^ 2 = \\ ={\varphi(q)\over q}\left\{\log{q\over 2\pi}+2\gamma+\sum_{p\mid q}{\log p\over p- 1}+\hbox{Re}{\Gamma'\over \Gamma}({1\over 2}+it) \right\}+2q^{- 1}\sum_{k\mid q}\mu(q/k)T({1\over 2}+it;k) \end{split} \] where \(k\) runs over all positive divisors of \(q\), \(p\) runs over all prime divisors of \(q\); and in the above formulae \(T(\sigma+it;k)\) has the following asymptotic expansion for any \(N>0\): \[ T(\sigma+it;k)=\hbox{Re }\sum^{N- 1}_{n=0}{-\sigma+it\choose n}k^{\sigma+it-n}\zeta(\sigma- n+it)\zeta(\sigma+n-it)+E_ N(\sigma+it;k), \] where \(E_ N(\sigma+it;k)\) is the error term satisfying the estimate \(E_ N(\sigma+it;k)=O(k^{\sigma-N})\) with the \(O\)-constant depending only on \(\sigma\), \(N\) and \(t\). In particular, if \(q=p\) is a prime, then we have the asymptotic expansions \[ \begin{split}(p-1)^{-1}\sum_{\chi\pmod p}| L(\sigma+it,\chi)|^ 2=\zeta(2\sigma)+p^{1- 2\sigma}\Gamma(2\sigma-1)\zeta(2\sigma-1)\times\\ \times \hbox{Re}\left\{{\Gamma(1- \sigma+it)\over\Gamma(\sigma+it)}\right\}+p^{-2\sigma }|\zeta(\sigma+it)|^ 2+2p^{-2\sigma}T(\sigma+it;p) \end{split} \] for \(\sigma\neq 1,2\) and \[ \begin{split} (p-1)^{- 1}\sum_{\chi\pmod p}| L({1\over 2}+it,\chi)|^ 2=\log{p\over 2\pi}+2\gamma+\hbox{Re }{\Gamma'\over \Gamma}({1\over 2}+it)-\\ -p^{- 1}|\zeta({1\over 2}+it)|^ 2+2p^{-1}T({1\over2}+it; p).\end{split} \] Further, in case \(N\geq2\), we obtain a more precise estimate for \(E_ N(\sigma+it;k)\) with respect to \(t\): \[ E_ N(\sigma+it;k)=O(k^{2\sigma}(| t|+1)^{1-2\sigma+2N}) \] for \(0<\sigma<1\) and any real \(t\), where the \(O\)-constant depends only on \(\sigma\) and \(N\).
For the proof, we use the method which has been developed by Y. Motohashi [Proc. Japan Acad., Ser. A 61, 222-224 (1985; Zbl 0573.10027)] and the above formulae are generalizations of an earlier result due to D. R. Heath-Brown [Comment. Math. Helv. 56, 148-161 (1981; Zbl 0457.10020)].
Reviewer: K.Matsumoto (Ueda)

MSC:

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

[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] Heath-Brown, D.R.: An asymptotic series for the mean value of DirichletL-functions. Comment. Math. Helv.56, 148–161 (1981) · Zbl 0457.10020 · doi:10.1007/BF02566206
[3] Motohashi, Y.: A note on the mean value of the zeta andL-functions I. Proc. Japan. Acad., Ser. A61, 222–224 (1985) · Zbl 0573.10027 · doi:10.3792/pjaa.61.222
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