arXiv:2203.05030v1 [math.FA] 9 Mar 2022 ORTHOGONALITY, DENSITY AND SHIFT-INVARIANCE IN THE HARDY SPACE APPROACH TO THE RH JUAN MANZUR, WALEED NOOR, AND CHARLES F. SANTOS Abstract. A Hardy space approach to the Nyman-Beurling and B´ aez-Duarte criterion for the Riemann Hypothesis (RH) was introduced in [16]. It states that the RH holds true if and only if the linear manifold N generated by a sequence of functions (h k ) k≥2 is dense in the Hardy space H 2 . In this article we ask the following three questions: A. How small is the orthogonal complement of N ? B. Is N dense in H 2 under a weaker topology? C. Is the closure of N a shift-invariant subspace of H 2 ? We use tools from local Dirichlet spaces, the de Branges-Rovnyak spaces and the Smirnov class to address these questions. In particular we prove that C is equivalent to the RH. Introduction The Riemann Hypothesis (RH) is considered the most important unsolved prob- lem in mathematics. In 1950, Nyman and Beurling gave functional analytic refor- mulation for the RH. Nyman’s thesis [17] contains a reformulation of RH in terms of density and approximation problems in L 2 (0, 1), which was extended to other L p spaces by Beurling [4] and refined five decades later by B´ aez-Duarte [2]. See the expository article [3] for details. Recently, the second author used tools from the Hardy-Hilbert space H 2 to further explore the Nyman-Beurling and B´aez-Duarte approach to the RH in [16] . For each k ≥ 2, define h k (z )= 1 1 − z log  1+ z + ··· + z k−1 k  and denote by N the linear span of {h k : k ≥ 2}. That each h k belongs to H 2 was proved in [16, Lemma 7]. One of the main results of [16] was the following reformulation of the RH as a completeness problem in H 2 . Theorem 1. (See [16, Theorem 8]) The RH holds if and only if N is dense in H 2 . This result raises the following three questions. A. Which H 2 functions (if any) could possibly be orthogonal to N ? B. Under which topologies (weaker than that of H 2 ) could N be dense in H 2 ? C. Is the closure of N a shift-invariant subspace? This goal of this article is to address these questions. For question A, since the RH is equivalent to N ⊥ = {0}, it is desirable to obtain linear subspaces V ⊂ H 2 as large as possible with N ⊥ ∩ V = {0}. By [16, Theorem 12] we can choose V to be the local Dirichlet space D δ1 which is dense in H 2 and contains all the functions Key words and phrases. Riemann hypothesis, Hardy space, Dirichlet space, de Branges- Rovnyak space, Smirnov class. 1