Arithmetical Aspects of Beurling’s Real Variable Reformulation of the Riemann Hypothesis

Let ρ(x) := x−[x], χ := χ(0,1], the characteristic function of (0, 1], λ(x) := χ(x) log x, and M(x) := ∑ k≤x μ(k), where μ is the Möbius function. B is the space of functions defined in (0,∞) by expressions ∑n k=1 ck ρ(θk/x) with n ∈ N, ck ∈ C and θk ∈ (0, 1]. A minor sharpening of the results of B. Nyman and A. Beurling states that for any fixed p ∈ (1,∞)B Lp the Riemann zeta function ζ(s) 6= 0 for Rs > 1/p, if and only if Lp(0, 1) ⊂ B Lp , which, furthermore, is equivalent to χ ∈ B Lp or to λ ∈ B Lp . Starting from the elementary identity λ(x) := ∫ 1 0 M1(θ)ρ(θ/x)θdθ, with M1(θ) := M(1/θ), where the integral suggests a limit of functions in B, we were led to the following two arithmetical versions of the Nyman-Beurling results, proved by classical, quasi elementary, number-theoretic methods. Define Gn, a natural approximation to λ, by Gn(x) := ∫ 1 1/n M1(θ)ρ(θ/x)θ −1dθ, then for all p ∈ (1,∞) (I) ‖Gn − λ‖p → 0 implies ζ(s) 6= 0 in Rs ≥ 1/p, and ζ(s) 6= 0 in Rs > 1/p implies ‖Gn − λ‖r → 0 for all r ∈ (1, p). Likewise noting that ζ(s) 6= 0 in Rs > 1/p is equivalent to ‖M1‖r < ∞ for all r ∈ (1, p), we have for all p ∈ (1,∞) (II) ‖M1‖p < ∞ implies λ ∈ B Lp , and λ ∈ B Lp implies ‖M1‖r < ∞ for all r ∈ (1, p). It is clear from (I) that Gn → λ diverges in L2, although it is shown to converge both pointwise and in L1 to λ. The general Lp case is also discussed. Some older natural approximations to χ, for which J. Lee, M. Balazard and E. Saias proved theorems analogous to (I), are shown to diverge in L2.