A generalization of the Riemann-Siegel formula

The celebrated Riemann-Siegel formula compares the Riemann zeta function on the critical line with its partial sums, expressing the difference between them as an expansion in terms of decreasing powers of the imaginary variable $t$. Siegel anticipated that this formula could be generalized to include the Hardy-Littlewood approximate functional equation, valid in any vertical strip, and this is our main result. We will discuss key ingredients: the saddle-point method, Mordell integrals, Bell polynomials and an interesting new family of polynomials that appear in the asymptotics. These polynomials inherit a functional equation from the Riemann zeta and they also seem to inherit interesting zeros from it too.

Cormac O'Sullivan