A simple principle can be taken as guide in extending basic results of analytic number theory as in Titchmarsh to algebraic number fields: Whenever you see a \(T\) in an estimate, with a logarithm, then replace it by \(d_kT^{N_k}\), where \(d_k\) is the discriminant of the field \(k\), and \(N_k\) is the degree. A systematic summary of the various basic estimates of analytic number theory is given, to show how the principle applies; including the Selberg estimate for the variation of argument of the zeta function up to the line \(1/2\). As an application one sees, using the Riemann Hypothesis, that the number of zeros of the zeta function of \(k\) in a box of height \(T\) fixed, but with variable \(k\) such that \(N_k/\log d_k\) goes to \(0\), is given asymptotically by \(\pi^{-1}T \log d_k\). Using the Hecke formula in a manner similar to that of Siegel's evaluation for the class number, one finds a necessary and sufficient condition for the zeta function of an imaginary quadratic field to have a zero at \(\frac12\) in terms of the class number. Conjecturably, this cannot happen in the imaginary quadratic case.