We give a complete classification of a certain family of step functions related to the Nyman–Beurling approach to the Riemann hypothesis and previously studied by Vasyunin. Equivalently, we completely describe when certain sequences of ratios of factorial products are always integral. Essentially, once certain observations have been made, this comes down to an application of Beukers and Heckman's classification of the monodromy of the hypergeometric function nFn−1. We also note applications to the classification of cyclic quotient singularities.

Abstract. In this paper we prove a discrete version of Tanaka's theorem [19] for the Hardy-Littlewood maximal operator in dimension n = 1, both in the non-centered and centered cases. For the non-centered maximal operator M we prove that, given a function f : Z → R of bounded variation,where Var(f ) represents the total variation of f . For the centered maximal operator M we prove that, given a function f : Z → R such that f ∈ ℓ 1 (Z),This provides a positive solution to a question of Haj lasz and Onninen [6] in the discrete one-dimensional case.

Let M (χ) denote the maximum of | n≤N χ(n)| for a given non-principal Dirichlet character χ (mod q), and let N χ denote a point at which the maximum is attained. In this article we study the distribution of M (χ)/ √ q as one varies over characters (mod q), where q is prime, and investigate the location of N χ . We show that the distribution of M (χ)/ √ q converges weakly to a universal distribution Φ, uniformly throughout most of the possible range, and get (doubly exponential decay) estimates for Φ's tail. Almost all χ for which M (χ) is large are odd characters that are 1-pretentious. Now, M (χ) ≥ | n≤q/2 χ(n)| = |2−χ(2)| π √ q|L(1, χ)|, and one knows how often the latter expression is large, which has been how earlier lower bounds on Φ were mostly proved. We show, though, that for most χ with M (χ) large, N χ is bounded away from q/2, and the value of M (χ) is little bit larger than √ q π |L(1, χ)|.

It is well-known that cancellation in short character sums (e.g. Burgess' estimates) yields bounds on the least quadratic nonresidue. Scant progress has been made on short character sums since Burgess' work, so it is desirable to find another approach to nonresidues. In this note we formulate a new line of attack on the least nonresidue via long character sums, an active area of research. Among other results, we demonstrate that improving the constant in the Pólya-Vinogradov inequality would lead to significant progress on nonresidues. Moreover, conditionally on a conjecture on long character sums, we show that the least nonresidue for any odd primitive character (mod k) is bounded by (log k) 1.4 .