PRH | Aux | 4.5 • Quantitative Beurling–Nyman from Helson–Blur

Quantitative Beurling–Nyman from Helson–Blur Hardy Approximation and an N−1/2Rate up to Logs Aleksandar Perišić August 2025 Abstract We derive a quantitative Beurling–Nyman (BN) approximation from the Helson–blur platform. On the Hardy side, we construct Dirichlet polynomials AN(s)with supp(AN)⊂ a dyadic prime block of size Nsuch that  (ζ(s)AN(s)−1)/s  H2(ℜs>1/2) ≪(log N)C √N. By the Mellin BN dictionary (see [1,2,3,4]) this yields, on the classical side, inf ak∈(0,1], ck∈C 1≤k≤N     1− N X k=1 ckρak    L2(0,1) ≪(log N)C √N, ρa(x) := na xo−an1 xo. In particular the BN distance tends to 0, recovering the criterion with a rate. All inputs (guarded window fit, vector balancing with oversampling, and moment/tail control) are imported from the Helson–blur papers [6,7]; we only perform the Hardy/Mellin conversion and an L2estimate with an optimal window height T∼N1/2. 1 Hardy set-up and what we import Let H2:= {Fholomorphic on {ℜs > 1/2}:∥F∥2 H2:= supσ>1/21 2πRR|F(σ+it)|2dt < ∞}. Classically (see [1,2,3,4]), Beurling–Nyman ⇐⇒ ζ·D contains 1in H2,(1.1) where Ddenotes Dirichlet polynomials Pnann−swith finite support (=Mellin image of the span of {ρa}; see §3). Imports from Helson–blur (no repetition). Fix rectangles K(T, δ) = {1/2 + δ≤ ℜs≤ 1,|t| ≤ T}. The companion work [6,7] constructs, for each Tand δ∈(0,1/2), a Dirichlet polynomial AT(s) = Pn∈STann−swith ST⊂(X, 2X]∩P,|ST|=: N, such that sup s∈K(T,δ)|ζ(s)AT(s)−1| ≤ αT,(1.2) αT≪sm(T) N·(log T)C, m(T)≍T, (1.3) N≫m(T) log m(T),(1.4) and the boundary guard excludes zeros on ∂K. Here m(T)is the size of a collocation grid on K(T, δ), and (1.3) is the vector-balancing residual (with oversampling (1.4)) from the core block. All constants depend at most on δ. 1 How the block polynomial ATis obtained. Although [6,7] focuses on the guard-and-fit mechanism rather than printing a single explicit closed-form polynomial, the construction is concrete: one fixes a collocation grid {sm}m(T) m=1 ⊂K(T, δ)(typically on a boundary mesh Γ(T, δ)), chooses a fresh dyadic prime block ST⊂(X, 2X]∩Pof size N, and forms the vectors vp:= ζ(sm)p−smm(T) m=1 ∈Cm(T)(p∈ ST). One then selects coefficients {ap}p∈STby a balancing/least-squares step so that Pp∈STapvp≈ (1,...,1) with residual αT, and defines AT(s) := Pp∈STapp−s. The boundary guard (no zeros on ∂K) upgrades the mesh fit to a uniform bound on K(T, δ), and oversampling (Nlarge compared to m(T)) forces the residual down to the scale in (1.3). Remark 1.1. We use only: (i) availability of a block-supported ATwith (1.2) and (1.3); (ii) |ST|=Nis tunable by enlarging the dyadic block; (iii) the guard allows Phragmén–Lindelöf transfers inside K(T, δ)(cf. [5, Ch. 5]). 2 From window fit to a global H2bound Write GT(s) := ζ(s)AT(s)−1. We estimate ∥(GT/s)∥H2by evaluating the H2-norm at σ= 1 (the supσ>1/2dominates any fixed σ): Lemma 2.1. For σ= 1 and every T≥2, 1 2πZR |GT(1 + it)|2 |1 + it|2dt ≪α2 T+1 T. Proof. Split at |t| ≤ Tand |t|> T. On |t| ≤ T, by (1.2) (on K(T, δ), then slid to σ= 1 via the guard and a maximum principle inside K) we have |GT(1 + it)| ≤ CαT. Hence Z|t|≤T|GT(1 + it)|2 1 + t2dt ≪α2 TZ|t|≤T dt 1 + t2≪α2 T. For |t|> T, use |GT(1 + it)| ≪ 1 + |ζ(1 + it)||AT(1 + it)|. In the Helson–blur construction the coefficients are column-normalized, so Pn∈ST|an|/n ≪1(dyadic support), while |ζ(1 + it)| ≪ log(2 + |t|). Thus |GT(1 + it)| ≪ log(2 + |t|)+1and Z|t|>T |GT(1 + it)|2 1 + t2dt ≪Z|t|>T log2(2 + |t|) 1 + t2dt ≪1 T. Combine the pieces and divide by 2π. Theorem 2.2 (Hardy approximation with rate).Let N=|ST|and assume (1.3)–(1.4). Then     ζ(·)AT(·)−1 ·   H2≪m(T) N1/2 (log T)C+T−1/2. In particular, choosing m(T)≍Tand N≍Tlog Tgives     ζ(·)AT(·)−1 ·   H2≪(log T)C √T. Reparameterizing by N(so T≍N/ log N) yields     ζ(·)AT(·)−1 ·   H2≪(log N)C √N. Proof. Insert αTfrom (1.3) into Lemma 2.1. With m(T)≍Tand N≍Tlog T, we have αT≪(T/N)1/2(log T)C≪(log T)C′/√T. The bounds follow. 2 3 The BN dictionary and the L2(0,1) rate Let Mdenote the Mellin transform M[f](s) = R1 0f(x)xs−1dx. The classical BN dictionary (Nyman [1], Beurling [2]; see also Baez–Duarte [3] and Nikolski [4, Ch. 13]) states:   1− N X k=1 ckρak  L2(0,1) =C    1−ζ(s)A(s) s   H2 ,(3.1) where A(s) = Pckas k(equivalently, a Dirichlet polynomial supported on a finite set), and C > 0 is the Mellin–Plancherel constant. Thus (1.1) is equivalent to the original BN criterion, and any H2bound for (ζA −1)/s transfers to an L2(0,1) bound for 1−Pckρak. Theorem 3.1 (Quantitative BN rate).There exist absolute constants C, c > 0such that for each integer N≥2there are parameters a1, . . . , aN∈(0,1] and coefficients c1, . . . , cN∈Cwith      1− N X k=1 ckρak    L2(0,1) ≤C(log N)c √N. In particular, the BN distance tends to 0at least as N−1/2+o(1). Proof. Apply Theorem 2.2 and the dictionary (3.1). The coefficients ck,akarise from the same balanced construction (identifying ATwith Pckas k); the polylog exponent cabsorbs the log T factors coming from the window grid and the oversampling condition (1.4). Remark 3.2 (Optimal choice of the window).The bound ∥(GT/s)∥H2≪αT+T−1/2exhibits the standard min-type optimization: since αT≪(T/N)1/2(log T)C, the choice T≍N/ log N balances the two terms and yields the N−1/2rate up to polylogarithms. 4 Consequences and comments •Theorem 3.1 shows not only 1∈span{ρa}but gives an explicit decay of the BN distance along a natural sparsity schedule (one dyadic prime block per step). •The log-loss depends on the collocation grid and can be improved by adjusting the guard/profile; nothing prevents pushing the exponent cdown with a more aggressive window design (e.g. refined boundary sampling or band-limited probes). •The coefficients can be taken block-sparse (supported on one dyadic block), which may be of independent numerical interest for constructive BN approximants. Acknowledgments. This note sits on the outputs (fit/guard and balancing residuals) of the Helson–blur papers [6,7]. No additional number-theoretic inputs are required beyond dyadic prime supply. References [1] B. Nyman, On some groups and semigroups of translations, Thesis, University of Uppsala, 1950. [2] A. Beurling, A closure problem related to the Riemann zeta-function, Proc. Natl. Acad. Sci. USA 41 (1955), 312–314. 3 [3] L. Baez-Duarte, A strengthening of the Nyman–Beurling criterion for the Riemann Hypothesis, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 14 (2003), 5–11. [4] N. K. Nikolski, Operators, Functions, and Systems: An Easy Reading. Vol. 1, AMS, 2001. [5] E. C. Titchmarsh (revised by D. R. Heath-Brown), The Theory of the Riemann ZetaFunction, 2nd ed., Oxford University Press, 1986. [6] A. Perišić, Hilbert–Pólya Realizations via Blur, manuscript, August 2025. [7] A. Perišić, Helson–Blur: Boundary Guards, Detectors, and Four-Flow Budgets, manuscript, September 2025. 4