On Ramachandra's method for the mean value problems of various \(L\)-functions
\textit{K. Ramachandra} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 1 (1974), 81-97 (1975; Zbl 0305.10036)] found a simple proof of the estimate NEWLINE\[NEWLINE {\sum_\chi}^*\int_{-T}^T|L(\sigma + \text{it},\chi)|^4 dt \ll \varphi(q)T(\log qT)^4\tag{1}NEWLINE\]NEWLINE for \(1/2 - 1/\log(qT) \leq \sigma \leq 1/2 + 1/\log(qT)\) where \(\varphi\) is the Euler function, \(q,T \geq 2\), and \({}^*\) denotes summation over all primitive characters modulo \(q\). The author expounds Ramachandra's method and uses it to prove the bound, analogous to (1), NEWLINE\[NEWLINE{\sum_\chi}^*\int_{-T}^T|L(f,\chi, \tfrac 12 + \text{it})|^2 dt \ll \varphi(q)T(\log qT). NEWLINE\]NEWLINE Here we have \(f \in S_k(N,\psi)\), where \(S_k(N,\psi)\) is a properly defined subspace of \(S_k(\Gamma_1(N))\), the space of all cusp forms of weight \(k (\geq 1)\) with respect to the congruence subgroup \(\Gamma_1(N)\) of SL\(_2(\mathbb Z)\), and \(L(f,\chi,s)\) is the ``twisted'' \(L\)-function associated with \(f \in S_k(\Gamma_1(N))\).NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].
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