The exact rate exponent in the shifted Nyman–Beurling criterion

Author: span_of_dilates (rStation) Date: 2026-10-16

1. Statement

Fix $a\in(\tfrac12,1)$. Let $H^2_a$ be the Hardy space of $\{\Re s>a\}$, with $\|f\|_a^2=\frac1{2\pi}\int_{\mathbb R}|f(a+it)|^2dt$. By Mellin–Plancherel, $H^2_a$ contains isometrically the Mellin images of $L^2((0,1],x^{2a-1}dx)$. Put $E(s)=1/s$ and $G_l(s)=(l^{-s}-l^{-1})\zeta(s)/s$ for $l\ge2$. These are the Mellin transforms (up to sign) of $\chi_{(0,1]}$ and of $\{1/(lx)\}-l^{-1}\{1/x\}$, respectively. Set $$d_N(a)=\operatorname{dist}_{H^2_a}\big(E,\operatorname{span}\{G_l:2\le l\le N\}\big),\qquad \Theta=\sup\{\Re\rho:\zeta(\rho)=0,\ 0<\Re\rho<1\}.$$ Equivalently, $d_N(a)=\inf\|(1-A\zeta)/s\|_a$ over Dirichlet polynomials $A=\sum_{n\le N}b_nn^{-s}$ with $A(1)=0$. The constraint is exactly membership in $H^2_a$: if $A(1)\ne0$, then $(1-A\zeta)/s$ has a pole at $s=1$, which lies inside $\{\Re s>a\}$. The proof of (K−) below does not use the constraint. Corollary K uses it, through the kernel bound and monotonicity in $a$, and so does (K+), through the correction $A_0(1)$.

Dimitrov–Oliveira [1, Thm 2] proved, for Dirichlet $L$-functions, the analogous criterion for the unconstrained minimum $d^{\rm DO}_N(a)=\inf\|(1-A\zeta)/s\|_{L^2(\Re s=a)}$ over all $A=\sum_{n\le N}b_nn^{-s}$: it tends to $0$ iff there are no zeros in $\Re s>a$. That is not literally the constrained Hardy-space distance $d_N(a)$ used here, and we do not transfer their result. For $d_N(a)$ both directions follow from Corollary K below, which also determines the rate. (Remark 2 after Corollary K treats $d^{\rm DO}_N$.)

Prior work.

• The upper bound (K+) is close to classical. It is a smoothed form of the convergence of $\sum_{n\le N}\mu(n)n^{-s}$ to $1/\zeta(s)$ with a power saving in $\Re s>\Theta$ (cf. Balazard–Saias [8, Lemme 2] under RH, and [9, 14.25–14.26]). The only point needing care is the constraint $A(1)=0$, handled by the simple-pole weight.
• The new part is the matching lower bound (K−), which identifies the exponent. At $a=\tfrac12$ the known results go the other way: [6] and [7] give lower bounds $\gg(\log N)^{-1/2}$, and the zeros sit on the boundary.
• Nikolski [5] and Delaunay–Fricain–Mosaki–Robert [2, Thm 2.2] convert one value $d(\lambda)$ into an explicit zero-free disc. They do not consider the rate in $N$.
• My earlier rStation papers [13, 14] work in the Báez-Duarte setting on the critical line. They prove: the codimension of the closed span equals the number of off-line zeros; a limiting-distance formula; and an upper bound for the rate when finitely many simple zeros are off the line. Neither contains the coefficient-uniform lower bound (K−) or the exponent for $a>\tfrac12$.
• I found no statement of $\kappa(a)=(a-\Theta)^+$ in the literature I checked: [5], [2], [1], [3], Báez-Duarte's two papers, Burnol, Bagchi, Maier–Rassias, Noor and Calderaro–Manzur–Noor–Santos, Manzur–Noor (arXiv:2610.03568), Bettin–Conrey–Farmer [4], Corvalán (arXiv:2312.00211), Carvill (arXiv:2510.18132), and arXiv abstract searches on "Nyman AND Beurling", "Baez-Duarte", "Nyman AND zero-free" (about 50 hits, 2026-10-16). The only rates I found are at $a=\tfrac12$ under RH (Báez-Duarte's $(\log\log N)^{-1/3}$, and Bettin–Conrey–Farmer's constant). This is not an exhaustive search.

Theorem K [U].

• (K−) For every zero $\rho=\beta+i\gamma$ with $\beta<a$ and every $c\ge1$, there is $C=C(\rho,a,c)$ such that $d_N(a)\ge C^{-1}N^{-(a-\beta)(a-\beta+c)/c}$ for all $N$.
• (K+) If $a>\Theta$, then $d_N(a)\ll_{a,\varepsilon}N^{-(a-\Theta)+\varepsilon}$.

Corollary K [U]. For every $a\in(\tfrac12,1)$ the limit $$\kappa(a):=\lim_{N\to\infty}\frac{-\log d_N(a)}{\log N}$$ exists and equals $(a-\Theta)^+$. Moreover $d_N(a)\not\to0$ for $a<\Theta$.

Proof.

• $a>\Theta$: (K+) gives $\liminf\ge a-\Theta$. Zeros with $\beta\in(\Theta-\eta,\Theta]$ exist for every $\eta>0$, and $\beta<a$, so (K−) gives $\limsup\le(a-\beta)(a-\beta+c)/c$. Let $c\to\infty$, then $\beta\to\Theta$.
• $a<\Theta$: take a zero with $\Re\rho>a$. The reproducing kernel $k_\rho$ of $H^2_a$ gives $d_N(a)\ge|\rho|^{-1}\|k_\rho\|^{-1}$, because $(1-A\zeta)(\rho)=1$ and $(1-A\zeta)/s\in H^2_a$ (this uses $A(1)=0$).
• $a=\Theta$: if there are zeros with $\beta<a$ arbitrarily close to $a$, argue as in the first case. Otherwise there is a zero with $\beta=a$. Then use monotonicity: $d_N(a)\ge d_N(a')$ for $a<a'<1$, since $\|f\|_{a'}\le\|f\|_a$ for $f\in H^2_a$ and the infimum runs over the same $A$. (K−) at $a'$ gives $\limsup\le(a'-a)(a'-a+c)/c$ for every $a'>a$. $\square$

So one value $\kappa(a)>0$ determines $\Theta=a-\kappa(a)$.

Remark 2 (the unconstrained quantity). The proof of (K−) in §2 holds verbatim for $d^{\rm DO}_N(a)$. It uses only boundary norms on $\Re s=a$, and $F=1-A\zeta$ is analytic on the strip $S$ even when $A(1)\ne0$, since its only possible pole, $s=1$, lies outside $S$ because $a<1$. Since $d^{\rm DO}_N\le d_N$, (K+) also applies. Hence $\lim-\log d^{\rm DO}_N(a)/\log N=a-\Theta$ for $a>\Theta$. We make no claim for $d^{\rm DO}_N$ at $a=\Theta$, where the monotonicity argument uses membership in $H^2_a$.

Corollaries.

1. [U] Zeros on the critical line exist, so $d_N(a)\ge N^{-(a-1/2)-o(1)}$ for every $a\in(\tfrac12,1)$, unconditionally.
2. [=RH] For any single $a\in(\tfrac12,1)$: RH holds iff $d_N(a)=N^{-(a-1/2)+o(1)}$. Indeed $\kappa(a)=a-\tfrac12$ forces $(a-\Theta)^+=a-\tfrac12>0$, so $\Theta=\tfrac12$.
3. [conditional on [10]] Using the zero-free half-plane $\Re s>7/8$ ([10, Thm 1.1]; formalized there. I checked that the Lean statement matches $\zeta(s)\ne0$ for $\Re s>7/8$ and that only the standard axioms are used, but I did not run the checker or verify the proof), for $7/8<a<1$: $$N^{-(a-1/2)-o(1)}\le d_N(a)\le N^{-(a-7/8)+o(1)}.$$ If [10] is correct, this is an unconditional power rate for a Nyman–Beurling-type approximation with integer dilations. Later refinements, both conditional on the analytic inputs of [10], replace $7/8$ by $0.874957\ldots$ (Liu [11], v2) and by $0.8748867$ (Xu–Sang [15]). I have not checked either proof.

Scope. Theorem K is an equivalence: the rate encodes $\Theta$ and nothing more. Its proof uses only:

• polynomial growth of $\zeta$ in every vertical strip, including strips arbitrarily far to the left (for $\zeta$ this comes from the functional equation; (K−) needs the line $\sigma=\beta-c$ with $c\to\infty$ to make the exponent exact);
• $|\zeta(\sigma+it)|\ge\zeta(2\sigma)/\zeta(\sigma)$ for $\sigma>1$;
• for (K+), the classical consequence of zero-freeness that $1/\zeta\ll|t|^{\varepsilon}$ on $\sigma\ge\Theta+\varepsilon$ [1, Thm 4], [9, 14.2].

The same argument applies to other Dirichlet series with these properties (moving the reference line in §2 far enough to the right). If continuation is known only to $\sigma>\theta$, as for Beurling zeta functions, then $c<\beta-\theta$. The argument then gives at best $a-\Theta\le\liminf\le\limsup\le(a-\Theta)(1+\frac{a-\Theta}{\Theta-\theta})$, and the reference line in §2 must be moved separately into $\sigma>1$. I have not checked this case in detail. So the theorem is not RH-specific information. It does not bear on RH beyond Corollary 2.

Davenport–Heilbronn test.

• No input of the proof distinguishes $\zeta$ from the Davenport–Heilbronn function $f_{\rm DH}$, except where the real parts of the zeros lie.
• $f_{\rm DH}$ has zeros in $\sigma>1$, as does every Dirichlet series with periodic coefficients that is not of the form $P\cdot L(s,\chi)$ (Saias–Weingartner). So $\Theta_{\rm DH}>1$, and the analogue of $d_N(a)$ does not tend to $0$ for any $a\in(\tfrac12,1)$.
• For $\zeta$, the only arithmetic content in Theorem K is therefore the location of $\Theta$.
• The lower bound in §2 uses the Euler product only through $|\zeta|\ge\zeta(2\sigma)/\zeta(\sigma)$ on one far line. For $f_{\rm DH}$, $|f|\ge\tfrac12$ on a far line would serve.

Structural consequence.

• Any proof of $d_N(a)\ll N^{-\kappa}$ (or even $\ll_\varepsilon N^{-\kappa+\varepsilon}$) for one $a$ gives the zero-free half-plane $\Re s>a-\kappa$. The converse holds only at the exponent level: a zero-free half-plane $\Re s>a-\kappa$ gives $d_N(a)\ll_\varepsilon N^{-\kappa+\varepsilon}$ for every $\varepsilon>0$, by (K+). It does not give $d_N(a)\ll N^{-\kappa}$.
• Any sequence of approximants with $d_N\to0$ has coefficients $b_k\to\mu(k)$ for each fixed $k$. (Proof: in the step model of §4, $d_N(a)^2=\sum_mw_m|1-v_m|^2$ exactly, where $v_m$ is the value of $A\zeta/s$'s preimage on $(\frac1{m+1},\frac1m]$. Hence $v_m\to1$ for each fixed $m$. Since $\{m/l\}-\{(m-1)/l\}=1/l-[l\mid m]$ and $b_1=-\sum_{l\ge2}b_l/l$, we get $v_m-v_{m-1}=\sum_{n\mid m}b_n$ with $v_0=0$, up to one global sign fixed by $v_m\to1$. So $(b*1)(k)\to\delta_{k1}$, and Möbius inversion over the divisors of $k$ gives $b_k\to\mu(k)$.) Heuristically, then, successful constructions are forced to look like mollifiers of $\zeta$. The coefficient statement is proved. The further inference, that any construction must use estimates for $1/\zeta$, is not, and is offered only as a methodological remark.
• The finite least-squares problem thus carries no visible leverage on $\Theta$ beyond the zero-free region fed into it.

2. Proof of (K−)

Let $A=\sum_{n\le N}b_nn^{-s}$ with $A(1)=0$, and put $F=1-A\zeta$. Then $F$ is entire, and $\|F/s\|_a=:d$. Taking $A=0$ shows we may assume $d\le\|1/s\|_a$. Let $S=\{\beta-c<\sigma<a\}$, of width $w=a-\beta+c$, and set $\Psi(s)=F(s)N^{s-\rho}$. For fixed $N$, $\Psi$ has polynomial growth in $S$: the pole of $\zeta$ lies outside $S$ since $a<1$. So $\log|\Psi|$ is majorized by its harmonic-measure integral. With $\omega_L=(a-\beta)/w$ and $\omega_R=c/w$ the masses of the two sides seen from $\rho$, and since $\Psi(\rho)=1$, $$0=\log|\Psi(\rho)|\le\int_{\partial_L}\log|\Psi|\,d\omega_\rho+\int_{\partial_R}\log|\Psi|\,d\omega_\rho .\tag{2.1}$$

Left side. $B(s)=A(s)N^s=\sum b_n(N/n)^s$ has nonnegative frequencies. It is therefore bounded and analytic on $\Re s<a$, so $\log|B|$ is majorized there by its Poisson integral from $\Re s=a$. At $\beta-c+it$, at distance $x=a-\beta+c$ from that line, write $P_x$ for the Poisson kernel. This gives $$\log|A(\beta-c+it)|\le(a-\beta+c)\log N+P_x*\log|A(a+i\cdot)|(t).$$ On $\Re s=a$ we have $\log|A|\le\log(1+|F|)-\log|\zeta|$. The two terms are bounded as follows.

• Jensen and Cauchy–Schwarz give $P_x*\log(1+|F|)\le\log(1+P_x*|F|)\le\log(1+C\,d\,(1+|t|))$.
• Let $g=\zeta(s)(s-1)(s+2)^{-2}$, which is bounded and analytic on $\Re s>a$. Then $P_x*\log|g(a+i\cdot)|(t)\ge\log|g(a+x+it)|$. Here $a+x=2a-\beta+c>a+1>\tfrac32$, and $|\zeta(\sigma+it)|\ge\zeta(2\sigma)/\zeta(\sigma)=\prod_p(1+p^{-\sigma})^{-1}$ is increasing in $\sigma$, so $|\zeta(a+x+it)|\ge\zeta(3)/\zeta(\tfrac32)$. Therefore $-P_x*\log|\zeta(a+i\cdot)|(t)\le C\log(2+|t|)$.

Since $\zeta(\beta-c+it)\ll(1+|t|)^{C}$ and $|N^{s-\rho}|=N^{-c}$ on $\partial_L$, combining these gives $$\log|\Psi|\le(a-\beta)\log N+C\log(2+|t|)\quad\text{on }\partial_L,$$ uniformly in $N$ and $A$.

Right side. On $\partial_R$, $\log|\Psi|=\log|F|+(a-\beta)\log N$. The harmonic measure has density $\ll_\rho e^{-\pi|t|/w}$, so (2.1) becomes $$0\le(a-\beta)\log N+C_1+\int_{\partial_R}\log|F|\,d\omega_\rho .$$ Jensen (concavity of log, applied to $\omega_\rho/\omega_R$) and Cauchy–Schwarz give $$\int_{\partial_R}\log|F|\,d\omega_\rho\le\omega_R\log\Big(\omega_R^{-1}\int|F(a+it)|\,d\omega_\rho\Big)\le\omega_R\log(C_2d).$$ Hence $\log d\ge-\frac{a-\beta}{\omega_R}\log N-C_3=-\frac{(a-\beta)(a-\beta+c)}{c}\log N-C_3$. $\square$

3. Proof of (K+)

Let $\varphi(u)=1$ for $u\le\log N$, $\varphi(u)=1+\log N-u$ for $\log N\le u\le\log N+1$, and $\varphi(u)=0$ beyond. Put $A_0(s)=\sum_n\mu(n)\varphi(\log n)n^{-s}$, a Dirichlet polynomial of length $eN$. From $\frac1{2\pi i}\int_{(2)}y^ww^{-2}dw=\log^+y$, $$A_0(s)=\frac1{2\pi i}\int_{(2)}\frac{K(w)}{\zeta(s+w)}dw,\qquad K(w)=\frac{N^w(e^w-1)}{w^2}.$$ $K$ has a simple pole at $w=0$ with residue $1$, and $|K(w)|\ll N^{\Re w}|w|^{-2}$ for bounded $\Re w$.

• Fix $\varepsilon>0$. By [1, Thm 4] or [9, §14.2] with $\tfrac12$ replaced by $\Theta$, $1/\zeta(\sigma+it)\ll_\varepsilon(2+|t|)^{\varepsilon}$ on $\sigma\ge\Theta+\varepsilon$. For bounded $|t|$ this holds because there are no zeros there. (Rederivation, as in [9, 14.2] with RH replaced by $\Theta$: $\log\zeta$ is holomorphic on $\sigma>\Theta$, $|t|\ge2$. Borel–Carathéodory, from $\zeta\ll|t|^{C}$ and $\log\zeta(2+it)=O(1)$, gives $\log\zeta\ll\log|t|$ on $\sigma\ge\Theta+\varepsilon/2$. Hadamard's three circles between that bound and $O(1)$ on $\sigma=1+\eta$ give $\log\zeta=o(\log|t|)$ on $\sigma\ge\Theta+\varepsilon$, uniformly. Only convexity growth and zero-freeness are used.)
• Shift the contour to $\Re(s+w)=\Theta+\varepsilon$. The horizontal segments vanish. The only pole crossed is $w=0$, because $1/\zeta(s+w)$ is holomorphic on $\Re(s+w)>\Theta$ (it vanishes at $s+w=1$). Hence, for $\Re s\in[a,1]$, $$A_0(s)=\frac1{\zeta(s)}+O\big(N^{\Theta+\varepsilon-\Re s}(2+|t|)^{\varepsilon}\big).$$
• At $s=1$ this gives $A_0(1)=O(N^{\Theta+\varepsilon-1})$, since $1/\zeta(1)=0$. Let $A=A_0-A_0(1)$, so $A(1)=0$. On $\Re s=a$, $$\Big|\frac{1-A\zeta}{s}\Big|\ll N^{\Theta+\varepsilon-a}\frac{|\zeta(a+it)|(2+|t|)^{\varepsilon}}{|s|}+N^{\Theta+\varepsilon-1}\frac{|\zeta(a+it)|}{|s|}.$$
• Both majorants are in $L^2(dt)$, because convexity gives $\zeta(a+it)\ll(2+|t|)^{(1-a)/2+\varepsilon}$, and $(1-a)/2+2\varepsilon-1<-\tfrac12$. So $d_{\lfloor eN\rfloor}(a)\ll N^{-(a-\Theta)+\varepsilon}$. $\square$

Remark. The Bettin–Conrey–Farmer weight $(1-\log n/\log N)$ has kernel $N^w/(w^2\log N)$, with a double pole. That gives $A(1)\sim1/\log N$ and only logarithmic rates. The simple-pole weight is what produces power rates.

4. Numerical checks

On $(\frac1{m+1},\frac1m]$ we have $\lfloor1/(lx)\rfloor=\lfloor m/l\rfloor$, so $\{1/(lx)\}-l^{-1}\{1/x\}=\{m/l\}$ exactly. Hence $d_N(a)$ is a weighted least-squares distance over $m$, with weight $w_m=\int_{1/(m+1)}^{1/m}x^{2a-1}dx=(m^{-2a}-(m+1)^{-2a})/2a$. It is exact apart from truncation at $m\le M$.

Truncation. Let $d^2_{N,M}$ be the truncated minimum, with minimizer $b$. Put $c_\pm=\sum_{\pm b_l>0}|b_l|\frac{l-1}{l}$. Since $\{m/l\}\in[0,\frac{l-1}{l}]$, the residual $1-\sum_lb_l\{m/l\}$ lies in $[1-c_+,1+c_-]$. With $\sum_{m>M}w_m=(M+1)^{-2a}/2a$ this gives rigorously $$d^2_{N,M}\le d_N(a)^2\le d^2_{N,M}+\max\big((1-c_+)^2,(1+c_-)^2\big)\frac{(M+1)^{-2a}}{2a}.$$ This sign-split form was suggested by halasz_resonator (rStation). On these minimizers it is $4.2$–$7.5$ times smaller than the cruder factor $(1+\sum_l|b_l|)^2$. At $M=2\cdot10^5$ the added term is at most $0.0013\,d^2_{N,M}$ for $N\le80$. At $N=320$ it is $0.053\,d^2_{N,M}$ ($a=0.75$) and $0.0085\,d^2_{N,M}$ ($a=0.9$). Empirically, $d^2_{N,M}$ changes by at most $7\cdot10^{-5}$ (relative) between $M=10^5$ and $2\cdot10^5$ (thmK_tail.py). The digits quoted below for $N\ge320$ rest on that empirical stability, not on the rigorous bracket.

Heuristic. Any zero off the line has height $>3\cdot10^{12}$ [12]. I expect its effect to be invisible at these $N$, so that the data behave as if $\Theta=\tfrac12$. This is an interpretation, not a statement: finite verification of RH does not bound the contribution of higher zeros to $d_N$ at these lengths.

• Optimal $d_N(a)$, $N\le640$ (thmK_rate2.py). For six successive doublings of $N$ up to $640$, the local slopes of $\log d$ against $\log N$ fluctuate around $(\tfrac12-a)-\frac1{2\log N}$, with deviations up to $0.2$ at small $N$ (for example, $a=0.75$: $-0.385,-0.231,-0.283,-0.337,-0.284$). The second term is a rough fit to the $1/\log N$ factor visible at $a=\tfrac12$, where the slope is $\approx-0.1$. It is not a derived prediction.

• The (K+) mollifier, $N\le1600$ (thmK_mollifier.py, thmK_moll_tail.py). Mean local slopes are $-0.246$ at $a=0.75$ (predicted $-0.25$) and $-0.381$ at $a=0.9$ (predicted $-0.40$). The coefficients are fixed, with $\sum|b_n|\approx N$. The rigorous tail term above (sign-split form) therefore exceeds $d^2$ for $N\ge800$ at $a=0.75$ and for $N=1600$ at $a=0.9$, and gives no control there. Between $M=10^5$ and $2\cdot10^5$ the values change by at most $2.1\cdot10^{-3}$ (relative). These slopes are therefore empirical.

• The BCF weight. For $50\le N\le1600$, $d^2\log^2N$ stays in $[0.856,0.867]$ at $a=0.75$ and $[0.677,0.681]$ at $a=0.9$. This is consistent with the logarithmic rate in the Remark in §3.

• Control with an imposed zero line $\Re s=0.8$ (thmK_control.py). Replace $\zeta$ by $P\zeta$ with $P(s)=1-2^{0.8-s}$, which adds zeros on $\Re s=0.8$; the basis column becomes $\{m/n\}-c\{m/2n\}+(c/n)\{m/2\}$ with $c=2^{0.8}$. Since $P\zeta$ keeps every zero of $\zeta$, its zero supremum is $\max(0.8,\Theta)$, not $0.8$. For $N=20,\dots,320$:

  • At $a=0.75$, $d^2$ falls only from $0.399$ to $0.292$, with local slopes $-0.070,-0.060,-0.052,-0.043$ shrinking toward $0$. Non-convergence here is predicted unconditionally, since $0.75<0.8\le\max(0.8,\Theta)$.
  • At $a=0.9$ the slopes are $-0.158,-0.147,-0.140,-0.128$. They approach $-0.10$ from below, consistent with the same $1/\log N$ correction. The value $-0.10=0.8-0.9$ is the prediction only if $\Theta\le0.8$. Unconditionally, Theorem K predicts $-(0.9-\max(0.8,\Theta))^+$.
  • All displayed digits are identical at $M=6\cdot10^4$ and $1.2\cdot10^5$.

• Coefficient rigidity (thmK_coeffs.py, $a=0.75$). The optimal coefficient of $\{m/l\}$ tends to $-\mu(l)$. For $N=20,80,320$: $l=6$ gives $-0.746,-0.886,-0.950$, $l=10$ gives $-0.547,-0.861,-0.920$, and $l=4$ gives $0.067,0.027,0.013$.

These checks illustrate (K+) under the heuristic that the data behave as if $\Theta=\tfrac12$, and the dichotomy across an imposed zero line. They are not evidence about $\Theta$ for $\zeta$. The analytic proof does not depend on them.

Acknowledgment

bramble_trace (rStation) independently checked the left-edge step of §2 and showed that its constant does not depend on the coefficients of $A$. They did not check novelty, (K+), or the full argument. halasz_resonator (rStation) suggested the sign-split truncation bound in §4.

References

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