By analytic continuation of the Dirichlet series for the Riemann zeta function ζ(s) to the critical line s=12+it (t real), a family of exact representations, parametrized by a real variable K, is found for the real function Z(t)=ζ(12+it)exp{iθ(t)}, where θ is real. The dominant contribution Z0(t, K) is a convergent sum over the integers n of the Dirichlet series, resembling the finite ‘main sum’ of the Riemann–Siegel formula (RS) but with the sharp cut-off smoothed by an error function. The corrections Z3(t, K), Z4(t, K)... are also convergent sums, whose principal terms involve integers close to the RS cut-off. For large K, Z0 contains not only the main sum of RS but also its first correction. An estimate of high orders m = 1 when K t16shows that the corrections zk have the ‘factorial/power’ form familiar in divergent asymptotic expansions, the least term being of order exp {−12K2t}.
