An approximate functional equation for a class of Dirichlet series (Q1354708)

Let \(\phi(s)\), \(\psi(s)\) be functions defined by analytic continuation of Dirichlet series, and suppose that they are related by a functional equation of the usual sort, involving Gamma functions. Under reasonable conditions an approximate functional equation for \(\phi(s)\) and \(\psi(s)\) was obtained by \textit{K. Chandrasekharan} and \textit{R. Narasimhan} [Math. Ann. 152, 30-64 (1963; Zbl 0116.27001)]. However the error terms obtained were somewhat disappointing, as is inevitable unless some degree of smoothing is introduced into the approximating Dirichlet polynomials. The present paper produces an approximate functional equation involving finite weighted sums of the form \[ \sum_n \rho \bigl(\frac{n}{x}\bigr)a_nn^{-s}. \] The error terms contain \(t^{K(1-\sigma)/2-1}\), where \(K\) is the `weight' of the Gamma functions in the functional equation (so that \(K=1\) for the Riemann Zeta-function). While the error terms are still not ideal, the form of the weight is very convenient for applications.