On the mean value of the generalized Dirichlet -functions
Rong Ma, Yuan Yi, Yulong Zhang (2010)
Czechoslovak Mathematical Journal
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Let be an integer, let denote a Dirichlet character modulo For any real number we define the generalized Dirichlet -functions where with and both real. They can be extended to all by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet -functions especially for and , and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.