The fourth residual moment \(Z_{q\le R_0}\): its \(q=1\) structure, its rank-1 pin, and a bound below \(N^3\log N\)

2026-09-11, corrected the same day (section 4 rewritten around a cited theorem; the first version carried a reversed density-exponent condition and an overstated zero-free-region saving, both recorded in section 4). Continues RANK3_CONDUCTOR_SUM.md, which bounded the \(U\)-side of UPPER_BOUND.md (23) at rank 1's order. This document treats the one component of (23) that had no estimate at all: the fourth residual moment over the small moduli, \[ Z_{q\le R_0}=\sum_{q\le R_0}Z_{(q)},\qquad Z_{(q)}=\sum_{a\bmod q}^*\int_{I_{q,a}}|R_{q,a}|^4, \] with \(Q=\lfloor\sqrt N/3\rfloor\), \(R_0=Q/L\), \(L=\log N\), the arcs \(I_{q,a}=\{\|\alpha-a/q\|\le\delta_q\}\), \(\delta_q=Q/(qN)\), and \(R_{q,a}=F_N-P_{q,a}\) as in UPPER_BOUND.md sections 3 and 6.

Result, stated first.

1. The exact \(q=1\) structure. \(Z_{(1)}=\int_{|\beta|\le Q/N}|D(\beta)|^4d\beta\) with \(D=F_N-K_N=\sum_{n\le N}(\Lambda(n)-1)\exp1(n\beta)\). Its full-circle version is a Goldbach-type mean square of order \(N^3\) (section 2) and is not a useful proxy; the arc-restricted quantity is controlled by the sequence \(\Lambda(n)-1\) summed over windows of length \(H=N/(2Q)\asymp\sqrt N\), by Gallagher's lemma.
2. The pin exists. By Cauchy-Schwarz against \(|K_N|^2\), \[ \boxed{\ Z_{(1)}\ \ge\ \frac{3\,U_1^2}{2N^3+N}\ \ge\ \frac{3}{2N^3}\big(T_N-O(N^2L)\big)^2(1+o(1)),\ } \tag{Z'} \] and the same for \(Z_{q\le R_0}\ge Z_{(1)}\). It is quadratic where (K') of RANK3_CONDUCTOR_SUM.md was linear. Its consequence: a bound \(Z_{q\le R_0}\ll N^{3-\eta}\) for a fixed \(\eta>0\) would establish \(\zeta(s)\ne0\) for \(\operatorname{Re}s>1-\eta/4\) (section 3).
3. The first bound below the trivial \(N^3L\). For every fixed \(A>0\), unconditionally, \[ \boxed{\ Z_{q\le R_0}\ \ll_A\ N^3L^{-A}.\ } \tag{Z} \] Moduli \(q\le L^C\) go through Gallagher's lemma and Koukoulopoulos's Theorem 1.1 on primes in short arithmetic progressions, applied at the interval lengths \(H_q=qN/(2Q)\), between \(\tfrac32\sqrt N\) and \(\tfrac32L^C\sqrt N\) (section 4). Moduli \(L^C<q\le R_0\) go through Vaughan's bound (V) and the disjointness of the arcs, exactly as (28) does above \(R_0\) (section 5).
4. The complete budget (23) is now priced (section 6): every component is bounded unconditionally, the total it yields is \(N^3L^{-A}+N^{13/5}L^6\), which is (1) again, and the two pins (K') and (Z') state what a stronger bound on any pinned component would establish: \(N^{3-\delta}\) on a pinned component gives a zero-free half-plane, and the target \(N^{2+\epsilon}\) gives RH.

Grade: derived, one route, with one theorem cited from its arXiv text and one classical lemma cited from memory (section 1), finite checks at \(q=1\) in section 7. Nothing here is evidence about the zeros of \(\zeta\) or of any \(L\)-function; the one statement involving zeros is the consequence of a hypothetical bound, in the form UPPER_BOUND.md section 1 already uses.

1. Inputs

• (23): \(E(N)\le32U_Q+8Z_Q+4I_Q+O(N^2L^3)\), with \(Z_Q=Z_{q\le R_0}+Z_{q>R_0}\) and \(Z_{q>R_0}\ll N^{13/5}L^6\) by (28).
• The arcs \(I_{q,a}\), \(q\le Q\), are pairwise disjoint (UPPER_BOUND.md section 3), so \(\sum_{q\le Q}\sum_a^*\int_{I_{q,a}}|F_N|^2\le\int_{\mathbb T}|F_N|^2=d_N\ll NL\), and \(\sum_{q,a}\int_{I_{q,a}}|P_{q,a}|^2\le N\sum_{q\le Q}\mu(q)^2/\phi(q)\ll NL\). Hence the trivial bound \(Z_Q\le\sup|R|^2\sum_{q,a}\int_{I_{q,a}}|R_{q,a}|^2\ll N^2\cdot NL=N^3L\).
• (V), valid on \(I_{q,a}\) for \(q\le R_0\) since \(\delta_q\le q^{-2}\) there: \(|F_N(\alpha)|\ll(Nq^{-1/2}+N^{4/5}+\sqrt{Nq})L^{5/2}\).
• Gallagher's lemma (Montgomery, Topics in Multiplicative Number Theory, LNM 227, Lemma 1.9; cited from memory, not re-read here): for \(0<\theta\le\tfrac12\) and any finitely supported \(a_n\), \[ \int_{-\theta}^{\theta}\Big|\sum_na_n\exp1(n\beta)\Big|^2d\beta \ \ll\ \theta^2\int_{-\infty}^{\infty}\Big|\sum_{x<n\le x+1/(2\theta)}a_n\Big|^2dx. \tag{G} \] Scaling check: \(a_n=1\) for \(n\le N\), \(\theta\ge1/N\), both sides \(\asymp N\).
• Koukoulopoulos, Theorem 1.1 (D. Koukoulopoulos, Primes in short arithmetic progressions, arXiv:1405.6592v2, read from the arXiv text in this session). With \[ E(y,h;q)=\max_{(a,q)=1}\Big|\sum_{\substack{y<p\le y+h\\ p\equiv a\ (q)}}\log p-\frac h{\phi(q)}\Big|, \] and (1.2) there the zero-density hypothesis \(\sum_{q\le Q_K}\sum^*_{\chi\bmod q}N(\sigma,T,\chi)\ll(Q_K^2T)^{c(1-\sigma)}\log^M(Q_KT)\): Assume that (1.2) holds for some \(c\in[2,4]\). Fix \(A\ge1\) and \(\epsilon\in(0,1/3]\). If \(x\ge h\ge1\) and \(1\le Q_K^2\le h/x^{1-2/c+\epsilon}\), then \[ \int_x^{2x}\sum_{q\le Q_K}E(y,h;q)\,dy\ \ll_{\epsilon,A}\ \frac{hx}{(\log x)^A}. \tag{K} \] The paper records that (1.2) is known with \(c=12/5+\epsilon\), \(M=14\) (Montgomery, Theorem 12.2, and Huxley 1975), so the range \(Q_K^2\le h/x^{1/6+\epsilon}\) is unconditional. It is also known with \(c=3\), \(M=9\) (Montgomery, Theorem 12.1), which gives the range \(Q_K^2\le h/x^{1/3+\epsilon}\); that weaker input already covers everything used below. His \(Q_K\) is a bound on moduli and is not UPPER_BOUND.md's arc parameter \(Q\).

2. The exact \(q=1\) structure

At \(q=1\), \(P_{1,1}=K_N\), \(R_{1,1}=D:=F_N-K_N=\sum_{n=1}^Nd(n)\exp1(n\beta)\), \(d(n)=\Lambda(n)-1\), and \[ Z_{(1)}=\int_{|\beta|\le Q/N}|D(\beta)|^4\,d\beta. \]

The full circle is the wrong object. \(\int_{\mathbb T}|D|^4=\sum_{m=2}^{2N}|c_m|^2\) with \(c_m=\sum_{n_1+n_2=m,\ n_i\le N}d(n_1)d(n_2)\). Expanding, \(c_m=G_N(m)-2S_N(m)+r_N(m)\), where \(G_N(m)=\sum_{n_1+n_2=m}\Lambda(n_1)\Lambda(n_2)\) is the Goldbach count truncated at \(N\), \(S_N(m)=\sum\Lambda(n_1)\) over the same pairs, and \(r_N(m)\) is their number. For \(m\le N\) this is \(c_m=\big(G_N(m)-m\mathfrak S(m)\big)+m\big(\mathfrak S(m)-1\big)-2\Delta(m-1)+O(1)\), and the middle term alone has square sum \(\asymp N^3\), because \(\mathfrak S(m)-1\) is \(-1\) on every odd \(m\) and does not average to zero over short ranges. So \(\int_{\mathbb T}|D|^4\asymp N^3\) unconditionally, measured at \(0.87N^3\) in section 7 and at \(1.53\,\phi(q)N^3\) for general \(q\) in RANK3_QUARTIC_TOOLS.md. On the full circle \(D\) still contains every major arc with \(q\ge2\); subtracting \(K_N\) removes only the one at \(0\). This is why a full-circle identity in the style of (D5) does not help \(Z\), as RANK3_ROUTE_D.md section 7 already recorded: the \(N^3\) is the \(q\ge2\) arcs, not a bookkeeping artifact.

The arc is a short-interval statement. By (G) with \(\theta=Q/N\), \(H:=1/(2\theta)=N/(2Q)\), and \(d(n)\) supported on \(1\le n\le N\), \[ \int_{|\beta|\le Q/N}|D|^2\ \ll\ \frac{Q^2}{N^2}\,V_H,\qquad V_H:=\int_{-H}^{N}\Big(\sum_{\substack{x<n\le x+H\\ n\le N}}d(n)\Big)^2dx, \tag{1} \] the mean square of the prime-counting error over windows of length \(H\asymp\tfrac32\sqrt N\), with the windows at the two ends truncated. This is the structure: the \(q=1\) arc of width \(Q/N\) sees primes at resolution \(\sqrt N\), and nothing pointwise. Partial summation on the arc gives only \(|D(\beta)|\le\max_t|\Delta(t)|(1+2\pi|\beta|N)\), which at \(|\beta|=Q/N\) is \(\asymp\sqrt N\max|\Delta|\), worse than the trivial \(2N\) unconditionally; the arc is too wide for the classical major-arc approximation and too close to \(0\) for (V). The mean square (1) is the object that can be bounded.

3. The pin, and what a stronger bound would establish

Lemma. For every \(N\), \(Z_{(1)}\ge3U_1^2/(2N^3+N)\).

Proof. \(U_1=\int_{|\beta|\le Q/N}|K_N|^2|D|^2\le\big(\int_{|\beta|\le Q/N}|K_N|^4\big)^{1/2}Z_{(1)}^{1/2}\) by Cauchy-Schwarz, and \(\int_{|\beta|\le Q/N}|K_N|^4\le\int_{\mathbb T}|K_N|^4 =\#\{n_1+n_2=n_3+n_4:\,1\le n_i\le N\}=(2N^3+N)/3\). \(\square\)

With (30), \(U_1\ge T_N-O(N^2L)\), this is (Z'). The same argument at any squarefree \(q\) gives \(Z_{(q)}\ge3\phi(q)\,U_{(q)}^2/(2N^3+N)\) (Cauchy-Schwarz on each arc, then on the sum over \(a\)), so also \(Z_{(2)}\ge\tfrac{3}{8N^3}(T_N-O(N^2L))^2\) by (K'). Only \(q=1\) is needed below.

Consequence. Suppose \(Z_{q\le R_0}\ll N^{3-\eta}\) for one fixed \(0<\eta<1\). Then \(Z_{(1)}\ll N^{3-\eta}\), so by (Z') \(T_N\ll N^{3-\eta/2}\), so \(\sum_{t\le N}\Delta(t)^2\ll N^{3-\eta/2}\), and by the dyadic Cauchy-Schwarz and Mellin argument of RANK3_CONDUCTOR_SUM.md section 5, \(\zeta(s)\ne0\) for \(\operatorname{Re}s>1-\eta/4\). So the \(Z\)-side of (23) has a rank-1 pin with the constant \(\eta/4\) in place of \(\delta/2\): a quadratic pin delivers half the exponent. Under RH the pin is far from the truth, \(Z_{(1)}\gg NL^{c}\) against a conjectural \(Z_{(1)}\ll N^{5/2}L^{2}\); the numbers in section 7 sit between the two, at \(Z_{(1)}/N^{5/2}\) falling from \(4\times10^{-4}\) to \(1\times10^{-4}\) over the ladder. The pin's content is what a successful stronger bound would carry with it, and it answers the question this document was asked: a bound \(Z_{q\le R_0}\ll N^{3-\eta}\) would be a theorem about the zeros of \(\zeta\), of exactly the stated width.

4. Upper bound for \(q\le L^C\): Gallagher plus Koukoulopoulos

Correction notice. The first version of this section claimed a short-interval variance bound for all \(H\ge N^{1/6+\epsilon}\) from Montgomery's Theorem 12.1, whose density exponent is \(3/(2-\sigma)\le3\). That is reversed: with exponent \(c\) the argument needs \(G^c\le N^{2-\epsilon}\) for \(G\asymp N/H\), i.e. \(H\ge N^{1-2/c+\epsilon}\), which is \(N^{1/3+\epsilon}\) for \(c=3\) and \(N^{1/6+\epsilon}\) only for Huxley's \(c=12/5\). It also claimed that a zero-free region of Littlewood width \(c\log\log T/\log T\) gives an arbitrary log-power saving. It does not: \(N^{-\eta/2}\) with that \(\eta\) is \((\log N)^{-c/2}\), one fixed power, which the \(\log^MN\) in the density estimate can outweigh. Both statements are withdrawn. The section now rests on (K), whose own inputs are the density estimate (1.2) and its author's proof, and it is applied only at the interval lengths that actually occur, \(H_q\in[\tfrac32\sqrt N,\tfrac32L^C\sqrt N]\). A third correction, same day: the passage from \(\theta\) to \(\psi\) in the corollary had claimed a pointwise polylogarithmic bound on the proper prime powers in a window, which is false for windows near \(\sqrt N\); it is replaced by the integrated bound \(\int P_h^2\le hB_N^2\), and the remainder in (5) now carries its harmonic-sum factor.

Fix \(C>0\) and \(1\le q\le L^C\) (non-squarefree \(q\) have \(P_{q,a}=0\) and are handled identically with the \(\mu(q)/\phi(q)\) term absent). Coefficients: \(R_{q,a}(\beta)=\sum_{n\le N}r_{q,a}(n)\exp1(n\beta)\), \(r_{q,a}(n)=\Lambda(n)\exp1(na/q)-\mu(q)/\phi(q)\). Then \[ Z_{(q)}\le\sum_a^\sup_{I_{q,a}}|R_{q,a}|^2\int_{I_{q,a}}|R_{q,a}|^2 \le4N^2\sum_a^\int_{|\beta|\le\delta_q}|R_{q,a}|^2 \ll N^2\delta_q^2\sum_a^\int\Big|\sum_{\substack{x<n\le x+H_q\\ n\le N}}r_{q,a}(n)\Big|^2dx, \tag{2} \] by (G) with \(H_q=1/(2\delta_q)=qN/(2Q)\), windows truncated to \(n\le N\) as in (1), \(x\) from \(-H_q\) to \(N\). Grouping \(n\) by residue class, with \(\psi_{x,H}(q,b)=\sum_{x<n\le x+H,\,n\le N,\ n\equiv b}\Lambda(n)\) and \(H'=H'(x)\) the truncated window length, and using \(\sum_b^\exp1(ab/q)=\mu(q)\), \[ \sum_{x<n\le x+H_q}r_{q,a}(n) =\sum_{b\bmod q}^\exp1(ab/q)\Big[\psi_{x,H_q}(q,b)-\frac{H_q'}{\phi(q)}\Big] +\sum_{\substack{b\bmod q\\(b,q)>1}}\exp1(ab/q)\psi_{x,H_q}(q,b)+O(1). \] The non-coprime classes contain only prime powers of primes dividing \(q\), at most \(\omega(q)\log_2N\) of them in any window, each weighted by at most \(L\), so that sum is \(O(\omega(q)L^2)=O(L^3)\). Extending the sum over \(a\) from the reduced residues to all residues mod \(q\) (every term is nonnegative) and applying Parseval on \(\mathbb Z/q\mathbb Z\), \[ \sum_a^\Big|\sum_{x<n\le x+H_q}r_{q,a}(n)\Big|^2 \le2q\sum_b^*\Big|\psi_{x,H_q}(q,b)-\frac{H_q'}{\phi(q)}\Big|^2+O(qL^6). \tag{3} \] (The first version wrote \(2\phi(q)\) here, which would need orthogonality over the reduced residues alone; that gives Ramanujan sums, not a delta. The factor \(q\) is the correct one and is harmless.)

Corollary of (K), in the shape needed. Let \(1\le q\le L^C\) and \(h\in[\tfrac32\sqrt N,\tfrac32L^C\sqrt N]\). For every fixed \(A\), \[ \int_{1}^{N-h}\sum_{b\bmod q}^*\Big(\psi(y{+}h;q,b)-\psi(y;q,b)-\frac h{\phi(q)}\Big)^2dy \ \ll_{A,C}\ q\,h^2N\,L^{-A}. \tag{4} \]

Proof. Write \(\theta\) for the prime-only count and \(P_h(y)=\sum_{y<p^k\le y+h,\ k\ge2}\log p\) for the proper-prime-power mass of the window, so that for every reduced \(b\), \(0\le[\psi(y{+}h;q,b)-\psi(y;q,b)]-[\theta(y{+}h;q,b)-\theta(y;q,b)]\le P_h(y)\). No pointwise bound on \(P_h\) is used: near \(y\asymp h\asymp\sqrt N\) a window holds about \(\sqrt h\) prime squares, not a power of \(\log\). Integrated it is small. Each proper prime power \(p^k\le N+h\) contributes \(\log p\) to \(P_h(y)\) exactly for \(y\in[p^k-h,p^k)\), a set of measure \(h\), so \(\int_{-h}^NP_h(y)\,dy\le hB_N\) with \(B_N:=\sum_{p^k\le N+h,\ k\ge2}\log p\ll\sqrt NL\), and \(P_h\le B_N\) pointwise, whence \[ \int_{-h}^{N}P_h(y)^2\,dy\ \le\ hB_N^2\ \ll\ hNL^2\ \ll\ h^2NL^{-A}, \] the last step because \(h\ge\tfrac32\sqrt N\). Since \(\sum_b^(\text{error})^2\le2\sum_b^(\theta\text{-error})^2+2\phi(q)P_h(y)^2\) and \(\phi(q)\le L^C\), the prime powers cost \(O(hNL^{C+2})=O(h^2NL^{-A})\) in total and it suffices to bound the \(\theta\)-errors. For \(y<h\) bound those trivially: the integrand is \(O(h^2L^2)\) on a range of length \(h\), contributing \(O(h^3L^2)=O(h^2N\cdot hL^2/N)\ll h^2NL^{-A}\) since \(h/N\ll N^{-1/2}L^C\). So it suffices to treat \(\theta\) on \(y\in[h,N-h]\), which we cover by dyadic blocks \([x,2x]\) with \(h\le x\le N\). On each block, \(\theta(y{+}h;q,b)-\theta(y;q,b)-h/\phi(q)\) is bounded in modulus by \(E(y,h;q)\) for every reduced \(b\), and also, trivially, by \(\theta(y{+}h)-\theta(y)+h\ll h\) (Brun-Titchmarsh, \(h\ge N^{1/3}\)). Hence \[ \sum_b^*\Big(\theta(y{+}h;q,b)-\theta(y;q,b)-\frac h{\phi(q)}\Big)^2 \le\phi(q)\cdot E(y,h;q)^2\ll\phi(q)\,h\,E(y,h;q). \] Apply (K) with \(Q_K=q\) (the sum over \(q'\le q\) has nonnegative terms and dominates the single term \(q'=q\)), \(c=3\), \(\epsilon=1/12\): the hypothesis \(Q_K^2=q^2\le h/x^{1/3+1/12}\) holds since \(q^2\le L^{2C}\) and \(h/x^{5/12}\ge\tfrac32\sqrt N/N^{5/12}=\tfrac32N^{1/12}\) for \(x\le N\), and \(x\ge h\) holds on every block used. So \(\int_x^{2x}\phi(q)hE(y,h;q)dy\ll_A\phi(q)h\cdot hx(\log x)^{-A}\), and summing over the \(O(L)\) dyadic blocks gives (4) with one more power of \(L\), absorbed by renaming \(A\). \(\square\)

The corollary is the theorem's \(L^1\) statement multiplied by the trivial pointwise bound; nothing sharper than (K) is used, and the ranges are exactly those of (K) with Ingham's exponent. With Huxley's \(c=12/5\) the same proof runs for \(h\ge N^{1/6+\epsilon}\), a range not needed here.

Insert (3) and (4) into (2), using \(N^2\delta_q^2=Q^2/q^2\) and \(H_q^2=q^2N^2/(4Q^2)\), and the trivial bound on the truncated windows at both ends exactly as in the proof above: \[ Z_{(q)}\ll_{A,C}\frac{Q^2}{q^2}\Big[q\cdot qH_q^2NL^{-A}+qNL^6\Big] =\tfrac14q^2N^3L^{-A}+\frac{Q^2NL^6}{q}, \] and summing over \(q\le L^C\), the remainder carrying the harmonic sum \(\sum_{q\le L^C}1/q\ll C\log L\), \[ \boxed{\ Z_{q\le L^C}\ \ll_{A,C}\ N^3L^{3C-A}+N^2L^{6}\log L.\ } \tag{5} \]

5. Upper bound for \(L^C<q\le R_0\): Vaughan plus disjointness

On \(I_{q,a}\) with \(q\le R_0\), (V) gives \(|F_N|\ll(NL^{-C/2}+N^{4/5}+N^{3/4})L^{5/2}\ll NL^{5/2-C/2}\), and \(|P_{q,a}|\le N/\phi(q)\le\zeta(2)N(1+\log q)/q\ll NL^{1-C}\), so \(\sup_{I_{q,a}}|R_{q,a}|^2\ll N^2L^{5-C}\) uniformly. Then, by the disjointness of the arcs (section 1), \[ Z_{L^C<q\le R_0}\le\max_{L^C<q\le R_0}\sup_{I_{q,a}}|R_{q,a}|^2\cdot\sum_{q,a}\int_{I_{q,a}}|R_{q,a}|^2 \ll N^2L^{5-C}\cdot NL, \] \[ \boxed{\ Z_{L^C<q\le R_0}\ \ll\ N^3L^{6-C}.\ } \tag{6} \] This is (28)'s argument with the cut at \(L^C\) instead of \(R_0\). What it cannot reach is \(q\) below \(L^C\), where (V) saves nothing, which is why section 4 is needed and why the two ranges meet at a power of \(\log\).

6. The result, and the complete budget

Take \(C=A+6\) in (6) and \(A\to A+3C\) in (5): for every fixed \(A\), \[ Z_{q\le R_0}\ll_AN^3L^{-A}+N^2L^6\log L\ll_AN^3L^{-A}, \] which is (Z). Together with (28) for \(q>R_0\): \(Z_Q\ll_AN^3L^{-A}+N^{13/5}L^6\).

The budget of UPPER_BOUND.md (23), every component now bounded unconditionally:

Consequences:

• (23) now yields (1). \(E(N)\ll_AN^3L^{-A}+N^{13/5}L^6\): the same order UPPER_BOUND.md section 5 reached with polylogarithmic arcs, now reached with square-root arcs and every component priced. No new total bound.
• What a stronger bound on a pinned component would establish. By (K') at \(q=2\) and (Z') at \(q=1\), a bound \(N^{3-\delta}\) on any of \(U_1\), \(U_{2\le q\le R_0}\), \(Z_{(1)}\), \(Z_{q\le R_0}\) would give \(\Theta\le1-\delta/2\) (the \(U\)-side) or \(\Theta\le1-\delta/4\) (the \(Z\)-side): a zero-free half-plane for \(\zeta\), which is not known. The two unpinned components, \(I_Q\) and \(Z_{q>R_0}\), are already at \(N^{13/5}\).
• The target through (23). Reaching \(N^{2+\epsilon}\) would prove RH (UPPER_BOUND.md section 1); reaching \(N^{3-\delta}\) would prove a zero-free strip, by the pins. Ranks 1 and 3 are one quantity seen in two moments, so a bound of either strength on either component carries the same consequence. This replaces RANK3_SCOPE.md section 4's rank ordering with an equivalence.
• The log powers. Sharpening \(L^{-A}\) to a sub-power saving in (K) and (Z) is a matter of replacing (SW) and the inputs of Theorem 1.1 by Vinogradov-Korobov-strength inputs throughout; it leaves both pins unchanged and is not attempted.

7. Finite checks at \(q=1\)

rank3_z_component_probe.py, results in results_rank3_z_component_probe.json, \(N\in\{10^3,5\times10^3,2\times10^4,6\times10^4\}\), numpy only:

• \(Z_{(1)}\) and \(U_1\) on grids of \(32N\) and \(64N\) points agree to relative \(10^{-3}\) or better; \(\int_{\mathbb T}|D|^4/N^3\) is \(0.82,\ 0.86,\ 0.86,\ 0.87\), the full circle at order \(N^3\) as section 2 says.
• The pin (Z') holds at every \(N\), with \(Z_{(1)}\) above \(3U_1^2/(2N^3+N)\) by factors \(1.8\times10^4\) to \(2.3\times10^5\): far from binding at these \(N\), as expected of a quadratic pin at scales where \(U_1\ll N^2L^c\).
• \(Z_{(1)}/N^3\) falls \(1.4\times10^{-5}\to4.9\times10^{-7}\) and \(Z_{(1)}/N^{5/2}\) falls \(4.3\times10^{-4}\to1.2\times10^{-4}\) across the ladder; the measured \(Z_{(1)}\) is below \(N^{5/2}\) at every \(N\) tried. The ratio \(Z_{(1)}/(N\,V_H)\) is \(3\times10^{-4}\) and falling: the sup bound \(|D|\le2N\) used in (2) is lossy by that much at these \(N\), which is where a sharper bound would have to look.

These check identities, an inequality and measured ratios at small \(N\). They test no asymptotic statement, and in particular (4) is not measured here.

8. Scope

This document is one exact lower bound (Cauchy-Schwarz), one reduction (Gallagher's lemma to short-interval mean squares in bounded-modulus progressions), one cited theorem applied in its stated range, and one two-line extension of (28). It establishes the first unconditional bound below \(N^3\log N\) for the fourth residual moment over the small moduli, at rank 1's order, together with the pin that states what a stronger bound would establish. It proves no power saving, claims none, and establishes nothing about the zeros of \(\zeta\) or of any \(L\)-function.