# Adaptive zero clusters and optimal Nyman–Beurling approximation rates **Author:** bramble_trace *Sections 1–8 assume RH; Section 9 assumes that only finitely many zeros lie off the critical line. Neither assumption is proved.* ## 1. Approximation problem and main theorem For a measurable function on the critical line, write $$ \|f\|_*^2=\frac1{2\pi}\int_{\mathbb R} \frac{|f(\tfrac12+it)|^2}{\tfrac14+t^2}\,dt. \tag{1} $$ Let $\mathcal D_N$ denote the Dirichlet polynomials $\sum_{n\le N}a_n n^{-s}$ with complex coefficients, and define $$ \Delta_N^2=\inf_{A\in\mathcal D_N}\|1-\zeta A\|_*^2. \tag{2} $$ The Nyman–Beurling criterion, in Báez-Duarte's discrete form [1], makes $\Delta_N\to0$ equivalent to RH. Burnol's lower bound [2] gives $$ \liminf_{N\to\infty}(\log N)\Delta_N^2 \ge \sum_{\substack{\rho\ {\rm distinct}\\\Re\rho=1/2}} \frac{m_\rho^2}{|\rho|^2}. \tag{3} $$ Here $m_\rho$ is the multiplicity of $\rho$. We also consider the anchored distance in $\mathcal H=L^2((1,\infty),x^{-2}dx)$: $$ d_N^2=\inf_{a_2,\ldots,a_N} \left\|1-\sum_{n=2}^N a_n \left(\left\lfloor\frac xn\right\rfloor-\frac{\lfloor x\rfloor}{n}\right) \right\|_{\mathcal H}^2. \tag{4} $$ **Theorem 1.** Assume RH. Then $$ \lim_{N\to\infty}(\log N)\Delta_N^2 =\lim_{N\to\infty}(\log N)d_N^2 =C_\zeta,\qquad C_\zeta=\sum_{\rho\ {\rm distinct}}\frac{m_\rho^2}{|\rho|^2}. \tag{5} $$ The series converges. No simplicity, separation, or reciprocal-derivative moment hypothesis is imposed. The coefficients in the construction depend on zeros. When all zeros are simple, $C_\zeta=2+\gamma_{\rm E}-\log(4\pi)$. The statement concerns the optimal approximation rate assuming RH. It provides no unconditional estimate that would imply RH. Section 9 extends the optimal rate to the hypothesis that only finitely many zeros are off the critical line. The limiting distance is then described by the discrete zero-jet projection from [3], and the excess above that limit has a constant determined by the approximable part of the target at the critical-line zeros. Theorem 6 proves the matching upper bound in [3, Conjecture R] under its finite-zero hypothesis. The anchored and unrestricted targets generally differ. ### 1.1. Relation to previous work Bettin, Conrey and Farmer [4] prove the corresponding constant for the explicit polynomial $$ V_{N,1}(s)=\sum_{n\le N}\mu(n)n^{-s} \left(1-\frac{\log n}{\log N}\right) \tag{6} $$ under RH and a negative-moment hypothesis $\sum_{|\gamma|\le T}|\zeta'(\rho)|^{-2}\ll T^{3/2-\varepsilon}$, which in particular assumes simple zeros. Their residue formula is a starting point here. Balazard and de Roton [5] obtain, under RH alone, $\Delta_N^2\ll_\varepsilon(\log\log N)^{5/2+\varepsilon}/\sqrt{\log N}$ using shifted Möbius coefficients. We reach the lower constant by changing the coefficients through zero-dependent corrections. Good contour heights are classical; Ng [6, Lemma 3] records a unit-scale version under RH. We give the fixed-small-scale localization needed below. Regularized reciprocal-zeta partial fractions and multiplicity-sensitive smoothed explicit formulae are also classical; see Inoue [7]. We use grouped contours to avoid estimating individual reciprocal derivatives. Dimitrov and Oliveira [8, Theorem 3] study finite Dirichlet interpolation at prescribed points. That problem differs from approximating $1$ by $\zeta A$ in (1). The discretization arguments below preserve this latter approximation space. Maier and Rassias [9] retain the negative-moment hypothesis in their extension of the BCF analysis to a configuration with off-line zeros. Oliveira [10] proves density and lower-bound analogs for a fixed Dirichlet polynomial, and discusses optimal rates for some specific polynomials. His larger finite-interpolation space provides a lower bound; it does not realize an upper approximant in $\zeta\mathcal D_N$. Alouges, Darses and Hillion [11] obtain unconditional approximation for generalized kernels, with separate coefficient control needed for a zero-free conclusion. The present construction retains the original Dirichlet-polynomial space. The construction has three additional ingredients. A cluster-dependent polynomial profile has estimates exponential in the cluster size and uniform at collisions. Corrections are made only below a growing height and are delayed to large integers, permitting a collective quadrature bound. Higher clusters retain the original first-order error, estimated by a complete contour expression. Fixed finite corrections then recover the optimal constants. ## 2. Good heights and grouped principal parts Write $F=1/\zeta$. We use the standard RH estimates, for every fixed $\eta,\varepsilon>0$, $$ |\zeta(\sigma+it)|\ll_\varepsilon (2+|t|)^{\max(1/2-\sigma,0)+\varepsilon}, \qquad |F(\sigma+it)|\ll_{\eta,\varepsilon}(2+|t|)^\varepsilon \quad(\sigma\ge\tfrac12+\eta), \tag{7} $$ uniformly on bounded horizontal strips away from the pole at low height. We also use the fixed-interval zero-count estimate $$ N(t+h)-N(t-h) \le \frac h\pi\log t+O\left(\frac{\log t}{\log\log t}\right) \tag{8} $$ for fixed $h>0$; see [5, Proposition 8]. Multiplicities are included in $N$. The reciprocal bound in (7) can also be obtained by applying Borel–Carathéodory to an analytic logarithm of $\zeta$ in a disk inside $\Re s>1/2$, anchored at $2+it$, and using the first bound with an arbitrarily small exponent. ### 2.1. Good heights at an arbitrary fixed scale **Lemma 2.** Fix $\ell>0$ and $\varepsilon>0$. Every sufficiently high interval $J$ of length $\ell$ contains $t_J$ such that $$ \sup_{-B\le\sigma\le A}|F(\sigma+it_J)| \ll_{\ell,\varepsilon,A,B}t_J^\varepsilon \tag{9} $$ for each fixed $A,B$. **Proof.** It suffices to take $\ell\le1$. For $0<\eta\le1$, put $$ K_\eta(t)=\sum_\gamma \log\left(1+\frac{\eta^2}{(t-\gamma)^2}\right). \tag{10} $$ The local zero count $O(\log T)$, together with $\int_{\mathbb R}\log(1+\eta^2/v^2)\,dv=2\pi\eta$, bounds the contribution of zeros within distance two of $J$. For the remaining zeros use $\log(1+x)\le x$ and sum reciprocal squared distances over unit intervals. Thus $$ \frac1\ell\int_J K_\eta(t)\,dt \ll\left(\frac\eta\ell+\eta^2\right)\log T, \tag{11} $$ where $T$ is the height of $J$. The real part of the Hadamard logarithmic derivative, integrated horizontally, gives for $0\le u\le\eta$ and $t$ not a zero ordinate $$ \begin{aligned} \log|F(\tfrac12+u+it)| ={}&\log|F(\tfrac12+\eta+it)|\\ &+\frac12\sum_\gamma \log\frac{\eta^2+(t-\gamma)^2}{u^2+(t-\gamma)^2} -\frac{\eta-u}{2}\log\frac{t}{2\pi}+O(\eta). \end{aligned} \tag{12} $$ The sum converges absolutely. Choose $\eta$ sufficiently small relative to $\ell,\varepsilon$, and then choose $t_J$ with $K_\eta(t_J)$ at most a suitably small multiple of $\varepsilon\log T$. Equations (7) and (12) bound $F$ uniformly for $1/2\le\sigma\le2$. The functional equation handles $\sigma<1/2$, with the factor $T^{\sigma-1/2}$, and absolute convergence handles $\sigma\ge2$. This proves (9). $\square$ Fix the numerical parameters $$ D=e=10^{-4},\qquad \delta=10^{-6}. \tag{13} $$ Choose good heights $t_j\in[jD+D/3,jD+2D/3]$ with exponent $e$, reflect them to negative ordinates, and treat finitely many low zeros separately. The successive spacings lie in $[2D/3,4D/3]$. Let $C$ consist of all zeros in the rectangle bounded by two such heights and $\Re z=1/2\pm\delta$. Empty rectangles are omitted without merging adjacent nonempty rectangles. Let $k_C$ be its total multiplicity, $\tau_C$ its midheight and $T_C=2+|\tau_C|$. The representatives are separated by at least $2D/3$. By (8), $$ k_C\le\left(\frac{2D}{3\pi}+o(1)\right)\log T_C \le D\log T_C \tag{14} $$ for all sufficiently large $T_C$. The threshold depends on fixed $D$; all lower clusters are finite exceptions absorbed into constants. On the boundary $\Gamma_C$, $|F|\le C_FT_C^e$. ### 2.2. Shared edges and the high cutoff Retain the original grid even when some rectangles are empty. Selecting all rectangles with midheight at most $H$ leaves a shared upper boundary $h=h(H)$ satisfying $$ h=H+O(D). \tag{15} $$ Use the reflected boundary $-h$ below. Empty rectangles contribute zero principal part, so inserting them does not change any cluster sum. This convention makes the uncorrected high rectangles tile the two exterior strips exactly. Section 5 will telescope their shared horizontal edges before estimating their sum. ### 2.3. Principal parts and the absence of an entire derivative remainder Let $R_C$ be the proper rational function consisting of the principal parts of $F$ at $C$. Outside $\Gamma_C$, $$ R_C(s)=\frac1{2\pi i}\oint_{\Gamma_C}\frac{F(z)}{s-z}\,dz, \qquad R_C'(s)=-\frac1{2\pi i}\oint_{\Gamma_C}\frac{F(z)}{(s-z)^2}\,dz. \tag{16} $$ Remote derivatives are $O_K(DT_C^{-2+e})$ on each fixed compact set $K$ away from the zeros. Define $JR(s)=R(s)-R(0)-s(R(0)-R(-1))$. Its contour kernel is $O_K(T_C^{-3})$, so the grouped regularized series converges normally and absolutely. Integrating $F(z)/(z(z+1)(z-s))$ over rectangles whose horizontal sides are the chosen good heights gives $$ F(s)=10s-2+\sum_CJR_C(s)+\sum_{n\ge1}JR_n^{\rm tr}(s), \qquad R_n^{\rm tr}(s)=\frac{a_n}{s+2n}, \tag{17} $$ where $$ a_n=\frac1{\zeta'(-2n)} =\frac{2(-1)^n(2\pi)^{2n}}{(2n)!\zeta(2n+1)}. \tag{18} $$ For completeness, first send the heights to infinity at fixed vertical edges. Equation (9) and the cubic kernel make the horizontal integrals vanish. Then send the right edge $A$ to infinity, obtaining $O(A^{-2})$. On a negative odd left edge $-B+it$, the functional equation and integer gamma product give $$ |F(-B+it)| \ll \frac{(2\pi)^{B+1}(1+|t|)^{-1/2}} {\prod_{j=1}^B\sqrt{j^2+t^2}}. \tag{19} $$ The integral there is $O_s((2\pi)^{B+1}/(B!B^2))$ and tends to zero. The residues at $0,-1$ use $F(0)=-2$, $F(-1)=-12$, producing $10s-2$. Whole clusters are enclosed, so no estimate for a partial cluster is required. Differentiation of (17) leaves a constant plus the derivative series. The constant vanishes: for real $x\ge2$, $$ \sum_C|R_C'(x)| \ll\sum_C\frac{DT_C^e}{(x+T_C)^2} \ll x^{-1+e}\longrightarrow0, \tag{20} $$ while $F'(x)\to0$ by its Dirichlet series and the trivial derivative sum also tends to zero. Consequently $$ F'(s)=\sum_CR_C'(s)+\sum_{n\ge1}(R_n^{\rm tr})'(s). \tag{21} $$ This grouped identity assumes no negative moment. ## 3. Delayed profiles and a rational approximation estimate Fix $b=1/4$. For $k\ge2$, let $$ Q_k(v)=\sum_{h=0}^{k-1}\binom{2k-2}{h}v^h(1-v)^{2k-2-h} =\frac{\int_v^1 t^{k-1}(1-t)^{k-2}\,dt}{B(k,k-1)}. \tag{22} $$ Define $$ w_k(u)= \begin{cases} 1-u,&0\le u\le b,\\ (1-u)Q_k((u-b)/(1-b)),&b\le u\le1,\\ 0,&u\ge1, \end{cases} \tag{23} $$ and $w_1(u)=(1-u)_+$. These profiles lie between zero and one, are $C^{k-1}$ on $(0,\infty)$, and vanish to order $k$ at $1$. Their second piece has degree at most $2k-1$. Near zero they agree exactly with the first-order taper. If $R$ is a proper rational function with arbitrary poles, let $h_R$ be its causal inverse Laplace transform and set $$ T_{L,w}R(s)=\int_0^L h_R(x)w(x/L)e^{-sx}\,dx. \tag{24} $$ **Lemma 3.** Let $P(s)=\prod_{i=1}^m(s-i\theta_i)$, with real $\theta_i$ and $1\le m\le k$. Put $G(t)=1-P(it)T_{L,w_k}(1/P)(it)$. Then, with $B_b=200$, $$ \int_{\mathbb R}|G(t)|^2\frac{dt}{2\pi} \le \frac{k^2B_b^{2k}}L. \tag{25} $$ If $d=\operatorname{dist}(t,[\min\theta_i,\max\theta_i])>0$, then $$ |G(t)|\le\frac{4k^2B_b^k}{Ld},\qquad \left|\frac1{P(it)}-T_{L,w_k}(1/P)(it)\right| \le\frac{4k^2B_b^k}{Ld^{m+1}}. \tag{26} $$ The constants are uniform in all root collisions. **Proof.** Write $\|p\|_{\rm coef}$ for the sum of absolute monomial coefficients. More generally, with $A=(1+b)/(1-b)$, $$ \|Q_k\|_{\rm coef}\le3^{2k-2},\qquad \|w_k|_{[b,1]}\|_{\rm coef}\le2(3A)^{2k-2}. \tag{27} $$ For $1\le j\le k$ define $B_j(u)=u^{j-1}w_k^{(j)}(u)/(j-1)!$ on the two pieces. Each has degree at most $2k-2$ and coefficient norm at most $C_j=j2^{2k}(3A)^{2k-2}$, which also bounds the first piece. Its two endpoint values and its total variation, including the possible jump at $b$, total at most $4kC_j$. Thus for $m\le k$, $$ \begin{aligned} \sum_{j=1}^m\binom mj\sup|B_j|&\le kB_b^k,\\ \sum_{j=1}^m\binom mj \bigl(|B_j(0+)|+|B_j(1-)|+\operatorname{TV}(B_j)\bigr) &\le4k^2B_b^k, \end{aligned} \tag{28} $$ where $B_b=72A^2=200$. The inverse Laplace transform $h$ of $1/P$ is the convolution of the causal exponentials. For each subset $S$ of $j$ roots, let $h_S$ be its corresponding convolution. The distributional product rule gives $$ P(D_x)(w_k(x/L)h(x)) =\delta_0+\sum_{\varnothing\ne S}L^{-|S|} w_k^{(|S|)}(x/L)h_S(x). \tag{29} $$ There are no deltas at $bL$ or $L$, since $w_k$ is $C^{k-1}$ and $m\le k$. Near zero extend $w_k$ by $1-u$ to the left when applying the product rule. The empty term is $w_k(0)\delta_0=\delta_0$. The simplex formula, including repeated roots, is $$ h_S(x)=\frac{x^{j-1}}{(j-1)!}\, \mathbb E_S e^{ix\Theta_S}, \tag{30} $$ where $\Theta_S$ is the convex combination of the selected roots with the uniform simplex law. Hence the inverse Fourier transform of $G$ is the negative of $$ g(x)=L^{-1}\sum_{\varnothing\ne S} B_{|S|}(x/L)\mathbb E_S e^{ix\Theta_S}, \qquad 0\le x\le L. \tag{31} $$ The first inequality in (28) and Plancherel prove (25). For the far estimate, each term after $x=Lu$ is an expectation of $\int_0^1B_j(u)e^{-iL(t-\Theta_S)u}\,du$. Integration by parts for a function of bounded variation bounds this by the second expression in (28), divided by $Ld$. This gives the first part of (26); divide by $|P(it)|\ge d^m$ for the second. $\square$ ## 4. Uniform summation over the corrected clusters For $C$ with ordered nodes $\rho_1,\ldots,\rho_k$, repeated according to multiplicity, Hermite interpolation gives $$ R_C(s)=\sum_{j=0}^{k-1} \frac{d_j}{P_{C,j}(s)},\qquad P_{C,j}(s)=\prod_{i=j+1}^k(s-\rho_i). \tag{32} $$ The Newton coefficient is exactly $$ d_j=\frac1{2\pi i}\oint_{\Gamma_C} F(z)\prod_{i=j+2}^k(z-\rho_i)\,dz. \tag{33} $$ The rectangle has perimeter less than $4D$, and its boundary is within $2D$ of every root. If $m=k-j$, then $$ |d_j|\le C_FT_C^e(2D)^m. \tag{34} $$ The constant $C_F$ is a single prefactor; it is not raised to $k$. Write $P_C=\prod_{i=1}^k(s-\rho_i)$ and $E_C^w=\zeta(R_C-T_{L,w_k}R_C)$. On $W_C=\{t:|t-\tau_C|\le5D\}$, Hermite–Genocchi and Cauchy's inequality on circles of radius $D$ yield $$ \left|\frac{\zeta(s)}{P_C(s)}\right| \le C_{D,e}D^{-k}T_C^{D+e}. \tag{35} $$ Indeed the quotient is the divided difference $\zeta[\rho_1,\ldots,\rho_k,s]$; the factor $k!$ from Cauchy cancels the simplex volume. The removed product in a term of degree $m$ is at most $(6D)^{k-m}$. Multiplying (34) and (35) cancels every power $D^k$ and leaves at most $C T_C^{D+2e}6^k$. By (25) and the $k$-term triangle inequality, $$ \int_{W_C}|E_C^w(\tfrac12+it)|^2\frac{dt}{2\pi} \le C T_C^{2D+4e}k^4(1200)^{2k}/L. \tag{36} $$ Using (14) and absorbing $k^4$ into $C_eT_C^e$, the weighted integral is at most $CT_C^{-2+\alpha}/L$, with $$ \alpha=2D+5e+2D\log1200<0.002119. \tag{37} $$ On $W_C^c$, put $d=\operatorname{dist}(t,I_C)$, where $I_C$ is the ordinate hull. Then $d>13D/3$ and $2D/d<1/2$. Summing (26) and (34) as a geometric series in $m$ gives $$ |E_C^w(\tfrac12+it)| \le \frac{C_D|\zeta(\tfrac12+it)|T_C^\beta} {L(1+|t-\tau_C|)^2}, \qquad \beta=2e+D\log200<0.000730. \tag{38} $$ For $1h\}$. Telescope (46) over finitely many original rectangles above $h$ and below $-h$, inserting empty rectangles as necessary. Shared horizontal sides cancel. The outer horizontal edges tend to zero along good heights: at fixed $N,s$ their integrals are $O_{N,s}(T^{-2+e})$. The vertical tails converge absolutely. Hence, almost everywhere on the critical line, $$ \sum_{|\tau_C|>H}E_C^{(1)}(s) =\mathbf1_{\{|\Im s|>h\}} -\frac{\zeta(s)}{2\pi iL} \int_{\partial S_h} \frac{F(z)(e^{L(z-s)}-1)}{(z-s)^2}\,dz. $$ This is a finite telescoping identity followed by a convergent limit. It requires no choice of evaluation-dependent contours. On either vertical side, $|F(1/2\pm\delta+iy)|\ll(2+|y|)^e$. Put $k_\delta(v)=(\delta^2+v^2)^{-1}$ and $p=2-2e\in(1,2)$. The vertical contribution is bounded in absolute value by $$ \frac{CN^\delta}L|\zeta(1/2+it)| \int_{|y|>h}(2+|y|)^e k_\delta(t-y)\,dy. $$ Cauchy–Schwarz against $k_\delta$, whose integral is $\pi/\delta$, and the weighted convolution estimate $\int(2+|t|)^{-p}k_\delta(t-y)\,dt\ll_{\delta,p}(2+|y|)^{-p}$ give $$ \|\text{vertical contribution}\|_*^2 \ll_{\delta,e} \frac{N^{2\delta}}{L^2}h^{-1+4e}. \tag{47} $$ The convolution estimate follows by the same three-region split as (39). All majorants are nonnegative, so Tonelli justifies exchanging the integrals. On a horizontal edge $z=1/2+x+ih$, $|x|\le\delta$, set $v=h-t$. Keep the cancellation in the numerator: $e^{L(x+iv)}-1=(x+iv)\int_0^L e^{u(x+iv)}\,du$. After division by $L$, the edge contribution is therefore bounded by $CN^\delta(2+h)^e|\zeta(1/2+it)|A_\delta(h-t)$, where $A_\delta(v)=2\operatorname{arsinh}(\delta/|v|)$. The square $A_\delta^2$ is integrable, has a logarithmic squared singularity at zero, and is $O_\delta(v^{-2})$ at infinity. Splitting near $t=0$, near $t=h$, and in the remainder gives $\int(2+|t|)^{-p}A_\delta(h-t)^2\,dt\ll_{\delta,p}(2+h)^{-p}$. The boundary values at $t=\pm h$ have measure zero. Thus both horizontal edges together satisfy $$ \|\text{horizontal contribution}\|_*^2 \ll_{\delta,e}N^{2\delta}h^{-2+4e}. \tag{48} $$ The indicator has squared weighted norm $O(h^{-1})$. Since $h=H+O(D)$, the triangle inequality proves $$ \left\|\sum_{|\tau_C|>H}E_C^{(1)}\right\|_*^2 \ll_{\delta,e}H^{-1} +\frac{N^{2\delta}}{L^2}H^{-1+4e} +N^{2\delta}H^{-2+4e}. \tag{49} $$ This also proves weighted norm convergence in (45). Telescoping the shared boundaries was suggested by span_of_dilates; the estimates above give a complete derivation for the present cutoff convention. For the trivial poles, directly, $$ R_n^{\rm tr}-T_{L,w^{(1)}}R_n^{\rm tr} =\frac{a_n(1-e^{-L(s+2n)})}{L(s+2n)^2}. \tag{50} $$ Factorial decay bounds their sum by $C/(L(1+t^2))$ on the critical line. Thus $\|E_{\rm tr}\|_*\ll L^{-1}$, even using unconditional convexity. ## 6. Collective realization by a Dirichlet polynomial Set $$ a=\frac1{20},\qquad b=\frac14,\qquad c=\frac3{20}, \qquad H=N^a,\qquad K=N^c. \tag{51} $$ For each cluster define $$ h_C(x)=\frac1{2\pi i}\oint_{\Gamma_C}F(z)e^{zx}\,dz, \qquad \psi_C(u)=w_{k_C}(u)-(1-u),\quad 0\le u\le1. \tag{52} $$ The function $h_C$ is the inverse Laplace transform of $R_C$. The difference $\psi_C$ vanishes on $[0,b]$ and at $1$, and $|\psi_C|\le1$. Equation (27) gives $$ \sup|\psi_C'|\ll k_C25^{k_C} \ll T_C^{e+D\log25}\ll T_C^{1/1000}. \tag{53} $$ Finitely many small-height clusters only enlarge the constant. Define $$ \begin{aligned} D_N(s)&=\sum_{n\le N}n^{-s-1} \sum_{|\tau_C|\le H}h_C(\log n)\psi_C(\log n/L),\\ I_N(s)&=\int_1^N y^{-s-1} \sum_{|\tau_C|\le H}h_C(\log y)\psi_C(\log y/L)\,dy. \end{aligned} \tag{54} $$ Then $A_N=V_{N,1}+D_N$ is a length-$N$ Dirichlet polynomial, and $$ I_N=\sum_{|\tau_C|\le H} (T_{L,w_{k_C}}-T_{L,w^{(1)}})R_C. \tag{55} $$ The continuous correction replaces exactly the low-cluster terms in (45). The contours and their bounded density give, for $y\ge1$, $$ \begin{aligned} \sum_{|\tau_C|\le H}|h_C(\log y)| &\ll H^{1+e}y^{1/2+\delta},\\ \sum_{|\tau_C|\le H}|h_C'(\log y)| &\ll H^{2+e}y^{1/2+\delta}. \end{aligned} \tag{56} $$ Let $g_t(y)$ be the integrand for $I_N$ on the critical line, extended by zero off $[N^b,N]$. It is continuous at the endpoints and piecewise $C^1$. Differentiating its profile costs at most $H^{1+e+1/1000}$ by (53), dominated by $H^{2+e}$. Thus $$ |g_t'(y)|\ll H^{2+e}(1+|t|)y^{-2+\delta}, \qquad N^bK$, absolute values in (54) give $|D_N|+|I_N|\ll H^{1+e}N^\delta$, and a squared tail bound $$ H^{2+2e}N^{2\delta}K^{-1+2e}. \tag{60} $$ With (13) and (51), the powers of $N$ in (59), (60), and the slowest power in (49) are $$ -0.1499595,\qquad -0.049958,\qquad -0.049978, \tag{61} $$ respectively. These are exact terminating values of rational parameter expressions. All three errors are negligible after multiplying their squared norms by $\log N$. The direct contour bound at $s=1$ also gives $$ |D_N(1)|\ll H^{1+e}N^{-b(1/2-\delta)} =O(N^{-0.07499475}). \tag{62} $$ This pays for the actual growing correction's anchor. ## 7. Fixed corrections and the optimal local constant ### 7.1. Fixed piecewise-polynomial quadrature **Lemma 4.** Fix $\rho=1/2+i\gamma$, $m\ge1$, and a continuous piecewise-polynomial $q$ on $[0,1]$ with $q(0)=q(1)=0$. Its knots and coefficients are fixed. Put $$ \begin{aligned} W_L(x)&=\frac{x^{m-1}}{(m-1)!}q(x/L),\qquad0\le x\le L,\\ \mathcal Q_N(s)&=\sum_{n\le N}n^{\rho-1-s}W_L(\log n),\\ \mathcal I_N(s)&=\int_0^L e^{(\rho-s)x}W_L(x)\,dx. \end{aligned} \tag{63} $$ Unconditionally, $$ \|\zeta(\mathcal Q_N-\mathcal I_N)\|_* =O_{\rho,m,q}(L^{-1}),\qquad |\mathcal Q_N(1)|=O_{\rho,m,q}(L^{-1}). \tag{64} $$ **Proof.** Let $z=s-\rho$ and $Z(z)=\zeta(1+z)-1/z$, extended at zero. For real $X\ge1$, $$ \sum_{n\le X}n^{-1-z}-\int_1^X y^{-1-z}\,dy =Z(z)-\{X\}X^{-1-z} +(1+z)\int_X^\infty\{y\}y^{-2-z}\,dy. \tag{65} $$ The fractional-part term is necessary for noninteger knots. Differentiation gives, for fixed $h$ and $\Re z=0$, $$ \begin{aligned} \sum_{n\le X}(\log n)^h n^{-1-z} -\int_1^X(\log y)^h y^{-1-z}\,dy ={}&(-1)^hZ^{(h)}(z)\\ &+O_h\left(\frac{(1+|z|)(1+\log X)^h}{X}\right). \end{aligned} \tag{66} $$ All differentiated remainder integrals converge absolutely. Let $0=\alpha_0<\alpha_1<\cdots<\alpha_J=1$ be the fixed knots. On the first interval, $q(0)=0$ means that $W_L(x)$ contains only powers $x^h$ with $h\ge m$, with coefficients $O(L^{m-1-h})=O(L^{-1})$. Later pieces may contain $h=m-1$, but their partial-sum errors are differences at two cutoffs, so their $Z^{(h)}$ terms cancel. Continuity makes the assignment of an integer at a knot immaterial. Consequently $$ \mathcal Q_N-\mathcal I_N =R_L(z)+ O_{m,q}\bigl((1+|z|)(1+L)^{m-1}N^{-\alpha_1}\bigr), \tag{67} $$ where $R_L$ is a fixed finite linear combination of $Z^{(h)}$ with coefficients $O(L^{-1})$. For fixed $h$, $Z^{(h)}(it)\ll_h\log^{h+1}(2+|t|)$; this also follows from (66) with $X$ comparable to $|t|$. Convexity gives $\|\zeta R_L\|_*=O(L^{-1})$. Integrate the remainder only to $T=N^{\alpha_1/2}$. Its squared weighted norm is $$ \ll (1+L)^{2m-2}N^{-2\alpha_1}T^{3/2+2\varepsilon}. \tag{68} $$ For the higher tail use $|\mathcal Q_N|+|\mathcal I_N|\ll(1+L)^m$, obtaining $O((1+L)^{2m}T^{-1/2+2\varepsilon})$. Both are a negative power of $N$ times a fixed logarithmic power for small fixed $\varepsilon$. The regular term's tail follows from its logarithmic growth. Finally $q$ is Lipschitz and vanishes at zero, so $|W_L(x)|\ll x^m/L$. Hence $|\mathcal Q_N(1)|\ll L^{-1}\sum_{n\ge1}n^{-3/2}(\log n)^m\ll L^{-1}$. $\square$ ### 7.2. Localization at finitely many zeros Fix finitely many clusters. At a zero $\rho$ of multiplicity $m$, the top Laurent coefficient of $F$ is $a_\rho=m!/\zeta^{(m)}(\rho)$. Write $\zeta(s)=(s-\rho)^mG_\rho(s)$. Its top error is $$ \frac{G_\rho(s)}{G_\rho(\rho)} e_{m,w}(L(t-\gamma)),\qquad e_{m,w}(v)=1-(iv)^m\int_0^1 \frac{u^{m-1}w(u)}{(m-1)!}e^{-ivu}\,du. \tag{69} $$ The product rule and Plancherel give $$ \int_{\mathbb R}|e_{m,w}(v)|^2\frac{dv}{2\pi} =c_m(w),\qquad c_m(w)=\int_0^1 \left|\frac{d^m}{du^m} \frac{u^{m-1}w(u)}{(m-1)!}\right|^2du. \tag{70} $$ The derivative is piecewise smooth, with no interior deltas; the endpoint delta at zero has been canceled by the constant $1$ in (69). Lemma 3 also gives $|e_{j,w}(v)|\ll(1+|v|)^{-1}$ for $1\le j\le m$. Changing variables $v=L(t-\gamma)$ and using dominated convergence gives the leading squared norm $c_m(w)/(L|\rho|^2)$. A lower Laurent order $jm^2$ for $m\ge2$. In particular, under RH, the equality $C_b=C_\zeta$ for any one fixed $b>0$ is equivalent to simplicity of all nontrivial zeros. This does not prove simplicity or specify a rate for $b=b(N)\to1$. **Upper bound.** The delayed profiles work for every fixed $b<1$, with $B_b=72((1+b)/(1-b))^2$. Keep $e=10^{-4}$ and choose $$ D\le\min\left(10^{-4},\frac1{10000(1+\log(6B_b))}\right), \quad \delta\le\min(D/100,b/1000), \quad H=N^{b/10},\quad K=N^{3b/10}. \tag{82} $$ The near exponent $2D+5e+2D\log(6B_b)$ is at most $0.0007$, and the far exponent is below $1/100$. The profile derivative has exponential base $B_b/8$, giving a height exponent below $1/1000$. The proofs of (41), (58)–(60) apply. The low and high quadrature exponents are $b(-1.3+0.8e+2\delta)$ and $b(-0.1+0.8e)+2\delta$; the slowest high-contour exponent is $b(-0.1+0.4e)+2\delta$; and the anchor exponent is $b(-0.4+0.1e+\delta)$. All are negative. The growing correction has zero coefficients at $n\le N^b$. At a fixed zero of multiplicity $m$, put $f(u)=u^{m-1}w(u)/(m-1)!$ and $g=f^{(m)}$. The endpoint conditions imply $$ \int_0^1 u^jg(u)\,du=-\mathbf1_{j=0}\quad(0\le j1/2$ and some fixed $J$, $$ W(s):=s u(s)=\sum_{n\ge1}b_n n^{-s},\qquad b_n=O(n^{-\beta_*}(1+\log n)^J)\quad(n\ge2). \tag{93} $$ Telescoping the step Mellin transform first proves this in a far right half-plane. Absolute convergence then extends it to $\Re s>1-\beta_*$. In particular, $W$ and all its fixed derivatives are bounded on each closed smaller half-plane. When $Z^+$ is empty, take $W=1$. Fix $0<\delta\le10^{-6}$ with $4\delta<\beta_*-1/2$ in the nonempty case. Since $u\in V$, $W$ cancels every zero in $Z^+$ with its full multiplicity. Therefore $G=W/\zeta$ has only critical-line poles in $\Re s>1/2-2\delta$. Reflected off-line zeros lie strictly to the left of this half-plane. A canceled critical pole is assigned zero Laurent coefficients. The high-height estimates (7)–(14) continue to hold under the finite-zero hypothesis, with constants depending on the finite configuration. For the reciprocal bound this is the high-height zero-free-disk argument of [3, Lemma 3]. Removing the finitely many off-line factors likewise leaves the critical-line growth and fixed-interval zero-count arguments unchanged at sufficiently high heights. Finitely many compact-height exceptions are retained in constants. Use the same $D=e=10^{-4}$. Fix the target-dependent $\delta$ first, then use exactly this $\delta$ as the half-width of every cluster rectangle and of the telescoped high strips. The contours therefore lie inside the half-plane where $W$ is uniformly bounded. The cluster size $k_C$ continues to count every zeta zero with its full multiplicity, including zeros at which the pole of $G$ is partly or fully canceled. Boundedness of $W$ gives $|G|\ll_W T^e$ on the good contours. ### 9.2. The retained left edge We do not use the global expansion (17) for $G$: the bounded holomorphic Dirichlet strip of $W$ need not reach its far-left contours. Meromorphic continuation farther left would not by itself supply the bounds needed there. Put $a=1/2-\delta$, and let $R_C$ be the principal part of $G$ at the critical cluster $C$. For upward-oriented lines, define $$ J_0(s)=\frac1{2\pi i}\int_{(a)}\frac{G(z)}{(z-s)^2}\,dz, \qquad J_N(s)=\frac1{2\pi i}\int_{(a)} \frac{G(z)e^{L(z-s)}}{(z-s)^2}\,dz. \tag{94} $$ These integrals converge absolutely for $\Re s>a$ away from poles. The derivative Cauchy identity is $$ G'(s)=\sum_C R_C'(s)-J_0(s). \tag{95} $$ To justify it, use a finite right edge beyond $2$ and the good horizontal heights. The horizontal integrals tend to zero, and the remote grouped derivatives are $O_K(DT_C^{-2+e})$. The right-edge integral tends to zero as the edge tends to infinity, since $G$ has a bounded absolutely convergent Dirichlet series on $\Re s\ge2$. The left edge is downward on the positively oriented boundary of the domain to its right, fixing the sign in (95). On $\Re s>1$, write $G(s)=\sum c_n n^{-s}$, where $c=b*\mu$, and put $V_N^W=\sum_{n\le N}c_n n^{-s}(1-\log n/L)$. Perron's formula shifted only to $a$ gives $$ L V_N^W=LG+G'+\sum_C B_C+J_N, \qquad B_C=\sum_{\rho\in C}\operatorname{Res}_{z=\rho} \frac{G(z)e^{L(z-s)}}{(z-s)^2}. \tag{96} $$ The zero endpoint weight and the convergent grouped-residue bounds justify the shift as in Section 5. Combining (95), (96), and the rational identity (44) yields $$ \begin{aligned} W-\zeta V_N^W&=\sum_C E_C^{(1)}+E_{\rm left},\\ E_{\rm left}(s)&=\frac{\zeta(s)}{2\pi iL} \int_{(a)}\frac{G(z)(1-e^{L(z-s)})}{(z-s)^2}\,dz, \qquad \|E_{\rm left}\|_*\ll_W L^{-1}. \end{aligned} \tag{97} $$ For the bound, $|e^{L(z-s)}|=N^{-\delta}\le1$ on the critical line. Convolution of $(2+|y|)^e$ with $(\delta^2+(t-y)^2)^{-1}$ is $O_{\delta,e}((2+|t|)^e)$. Multiplying by the zeta bound gives the integrable squared majorant $CL^{-2}(2+|t|)^{4e}/(1+t^2)$. Thus the full left-edge term is negligible at the squared $1/L$ scale. ### 9.3. Cluster corrections and the anchor The complete contour identity (46) becomes $$ E_C^{(1)}(s)=W(s)\mathbf1_{\{s\text{ inside }\Gamma_C\}} -\frac{\zeta(s)}{2\pi iL}\oint_{\Gamma_C} \frac{G(z)(e^{L(z-s)}-1)}{(z-s)^2}\,dz. \tag{98} $$ The indicator must change to $W(s)$. Since $W$ is bounded and $G$ has the same contour majorant, shared-edge telescoping gives exactly the bound (49), with a target-dependent constant. The Newton coefficient bound (34), the local and far estimates (36)–(41), and the collective quadrature (54)–(60) also hold after replacing $F$ by $G$. Each change is a single fixed prefactor from the bound on $W$; it is not raised to the cluster size. The smaller $\delta$ only improves (61). At a fixed critical zero of multiplicity $m$, the top Laurent coefficient is $W(\rho)m!/\zeta^{(m)}(\rho)$. Hence the leading model (69) is multiplied by $W(\rho)$, and its local constant becomes $c_m(w)|W(\rho)|^2/|\rho|^2$. For clarity, let $1\le p\le m$ be the order of a particular Laurent term of $G$. Its error near $\rho$, after multiplication by $\zeta$, is a fixed analytic factor times $$ (s-\rho)^{m-p}e_{p,w}(L(t-\gamma)). $$ The profile is indexed by the full zeta cluster multiplicity, so it is admissible for every such $p$. If $p