The Selberg \(Z\)-function. Local approach.
The Selberg \(Z\)-function is defined by \[ Z_{m,n}(s)=\sum_{c=1}^\infty\frac{S(m, n; c)}{c^{2s}},\quad\Re s>3/4, \] where \(S(m, n; c)\) is the Kloosterman sum. \textit{A. Selberg} [Proc. Sympos. Pure Math. 8, 1--15 (1965; Zbl 0142.33903)] extended this function to the half-plane \(\Re s\leq3/4\). \textit{D. Goldfeld} and \textit{P. Sarnak} [Invent. Math. 71, 243--250 (1983; Zbl 0507.10029)] obtained upper bounds for \(Z\)-function in the critical strip. Here the author reproves the above results in a different way. For this he uses some local variant of the Selberg \(Z\)-function.
scientific article; zbMATH DE number 1759026