Inversion formula of Dirichlet polynomials and the approximate functional equation of Dirichlet's \(L\)-functions (Q1383616)

Let \(\chi\) be a non-trivial character modulo \(q\) with `Führer' \(q^*\), let \(g\) be an infinitely differentiable function with support contained in \([\frac 12,2]\) and fulfilling some more conditions. Then for \({\mathbb C}\ni s=\sigma+it\) with \(0<\sigma<1\), \(t>1\), \(N>0\) we have \[ \sum_{n} g(n/N)\chi(n)n^{-s} = A(s,\chi)\sum_{n}g(qt/(2\pi nN)) G(n,\chi)n^{s-1} + R, \] where \(A(s,\chi)\) denotes a well-known meromorphic function and \(G(n,\chi)\) the Gaussian sum. \(R\) is an explicit \(O\)-error term which contains the modulus \(q^*\). Using this result and the upper bound method of Heath-Brown, the author proves a smoothly weighted version of the approximate functional equation for Dirichlet's \(L\)-functions \(L(s,\chi)\) for \(\sigma \geq\frac 12\). The main progress lies in the fact that in the error term the non-trivial estimation \((qt)^{\varepsilon}(q^*t)^{3/16}\) occurs.