Orthogonal polynomial expansions for the Riemann xi function Dan Romik Author address: Department of Mathematics, University of California, Davis, One Shields Ave, Davis CA 95616, USA E-mail address:
[email protected] Contents Chapter 1. Introduction 1 1.1. Background 1 1.2. Our new results: Tur´an'sprogram revisited and extended; expansion of Ξ(t) in new orthogonal polynomial bases 3 1.3. Previous work involving the polynomials fn 5 1.4. How to read this paper 6 1.5. Acknowledgements 6 Chapter 2. The Hermite expansion of Ξ(t) 7 2.1. The basic convergence result for the Hermite expansion 7 2.2. Preliminaries 8 2.3. Proof of Theorem 2.1 8 2.4. An asymptotic formula for the coefficients b2n 12 2.5. The Poisson flow, P´olya-De Bruijn flow and the De Bruijn-Newman constant 17 Chapter 3. Expansion of Ξ(t) in the polynomials fn 20 3.1. Main results 20 3.2. Proof of Theorem 3.1 21 3.3. Proof of Theorem 3.2 23 3.4. The Poisson flow associated with the fn-expansion 26 3.5. Evolution of the zeros under the Poisson flow 27 Chapter 4. Radial Fourier self-transforms 30 4.1. Radial Fourier self-transforms on Rd and their construction from balanced functions 30 4.2. The radial function A(r) associated to !(x) 31 4.3. An orthonormal basis for radial self-transforms 33 4.4. Constructing new balanced functions from old 35 4.5. The functions ν(x) and B(r) 35 4.6.