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Library · hunts/prime_pair_error/RANK3_CONDUCTOR_SUM.md

Rank 3's mixed moment over the complete modulus range, organized by conductor

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2026-09-11. This document finishes the estimate that RANK3_COMPOSITE_CROSS_TERM.md section 5 left as its one remaining step: carry the exact conductor-weighted identity (CT0) over the complete range \(2\le q\le R_0\), and read off an unconditional bound for the rank-3 mixed moment \(\sum_{2\le q\le R_0}U_{(q)}\) of UPPER_BOUND.md (23). Along the way it corrects the Barban-Davenport-Halberstam statement that RANK3_MEAN_VALUE_TOOLS.md section 1 records and RANK3_POLYRANGE_TINT_CHECK.md section 3 uses, because as written that statement has no Siegel-Walfisz floor and its \(Q=1\) case is stronger than what RH would give.

Result, stated first. For every fixed \(A>0\), unconditionally, \[ \boxed{\ \sum_{q=2}^{R_0}U_{(q)} \ \le\ 4(C_3-1)\,T_N \ +\ O_A\!\big(N^3L^{-A}\big) \ +\ O\!\big(N^2L^2(\log\log N)^2\big), \qquad C_3=\prod_p\Big(1+\frac1{(p-1)^3}\Big)=2.3009\ldots\ } \tag{K} \] where \(T_N\) is the \(q=1\) quantity of UPPER_BOUND.md (29) and \(L=\log N\). Since \(T_N\ll_HN^3L^{-2H}\) by (SW) at \(q=1\) (UPPER_BOUND.md section 7), this gives \(\sum_{2\le q\le R_0}U_{(q)}\ll_AN^3L^{-A}+N^2L^{2+o(1)}\) for every fixed \(A\). In the other direction, the single modulus \(q=2\) already gives \[ \sum_{q=2}^{R_0}U_{(q)}\ \ge\ U_{(2)}\ \ge\ \tfrac12T_N-O(N^2L). \tag{K'} \] So the rank-3 mixed moment is pinned two-sidedly to rank 1's own quantity: its bound is the same order as rank 1's, and any fixed power saving on it, \(\sum_{2\le q\le R_0}U_{(q)}\ll N^{3-\delta}\), forces \(\zeta(s)\ne0\) for \(\operatorname{Re}s>1-\delta/2\) (section 5). The composite-modulus cancellation of RANK3_COMPOSITE_CROSS_TERM.md is used in full and does exactly what it can: it makes the character part of the sum smaller than the principal part, which is the part no averaging over \(q\) can remove. Section 6 records what this does to the budget (23): the whole \(U\)-side is now bounded at rank 1's order, and the only component of (23) without any estimate is the fourth moment \(Z_{q\le R_0}\).

Everything here is derived, one route, with a finite check in section 7. Nothing here is evidence about the zeros of \(\zeta\) or of any \(L\)-function; the one implication involving zeros (section 5) is a deduction from a hypothetical bound, in the same form UPPER_BOUND.md section 1 already states for the total.

1. Inputs, all already on this branch or classical

Notation is RANK3_ROUTE_D.md's and RANK3_CROSS_TERM_CANCELLATION.md's: \(Q=\lfloor\sqrt N/3\rfloor\), \(R_0=Q/L\), \(U_{(q)}\), \(T_N(q,a)\), \(R^{(1)}{q,a},R^{(2)}{q,a}\), \(\rho_2(q)\), \(\Sigma_{\rm diag}(q)\), \(\Sigma_{\rm cross}(q)\), \(E(q)\), and for a character \(\chi\bmod q\), \(\psi(t,\chi)=\sum_{n\le t}\Lambda(n)\chi(n)\) and \(M_\chi=\int_{\mathbb T}|K_NL(\chi)|^2\). By the same Parseval computation as (D5), for every character, \[ M_\chi=\sum_{t=1}^N|\psi(t,\chi)|^2+\sum_{t=1}^{N-1}|\psi(N,\chi)-\psi(t,\chi)|^2 \ \le\ 3\sum_{t=1}^N|\psi(t,\chi)|^2+2N|\psi(N,\chi)|^2. \tag{1} \]

2. The Barban-Davenport-Halberstam range, corrected

RANK3_MEAN_VALUE_TOOLS.md section 1 records (BDH) as \(D(x,Q)=\sum_{q\le Q}\sum_a^*(\psi(x;q,a)-x/\phi(q))^2\ll Qx\log x\) "uniformly for \(1\le Q\le x\)", and RANK3_POLYRANGE_TINT_CHECK.md section 3 applies it at every \(Q\) from \(\lfloor L^B\rfloor\) upward. At \(Q=1\) that statement reads \((\psi(x)-x)^2\ll x\log x\), which is stronger than the \(x\log^4x\) that RH itself gives (UPPER_BOUND.md section 7), so it cannot be the unconditional theorem. The theorem (Davenport, Chapter 29) is stated for \(x(\log x)^{-A}\le Q\le x\); below that range it carries a floor. The floor is visible from the character decomposition, which is also the computation this document needs, so here it is in full.

By orthogonality, \(\psi(x;q,a)=\phi(q)^{-1}\sum_\chi\bar\chi(a)\psi(x,\chi)\), and \(\sum_a^\bar\chi(a)=0\) for \(\chi\ne\chi_0\), so for every \(q\) \[ \sum_{a\bmod q}^\Big(\psi(x;q,a)-\frac x{\phi(q)}\Big)^2 =\frac{(\psi(x,\chi_0)-x)^2}{\phi(q)}+\frac1{\phi(q)}\sum_{\chi\ne\chi_0}|\psi(x,\chi)|^2. \tag{4} \] The first term is the principal contribution: \(\psi(x,\chi_0)-x= (\psi(x)-x)-r_q(x)\) with \(0\le r_q(x)\le\rho_2(q)\), so it is \((\psi(x)-x)^2/\phi(q)\) up to \(O(\rho_2(q)(|\psi(x)-x|+\rho_2(q))/\phi(q))\). It is present at every modulus and it does not shrink as \(q\) grows. Summed over \(q\le Q\) with the weight \(1/\phi(q)\) it contributes \((\psi(x)-x)^2\log(2Q)\), and unconditionally that is \(\ll_Ax^2(\log x)^{-A}\log(2Q)\) and nothing better without a zero-free strip.

Each nonprincipal \(\chi\bmod q\) is induced by a unique primitive \(\chi^\bmod q^\), \(q^\mid q\), \(q^\ge2\), and \(\psi(x,\chi)=\psi(x,\chi^)-\sum_{n\le x,(n,q)>1}\Lambda(n)\chi^(n)\), the correction being at most \(\rho_2(q)\) in modulus. Organizing the second term of (4) by conductor and summing over \(q\le Q\), the weight on \(|\psi(x,\chi^)|^2\) is \(\sum_{q\le Q,\,q^\mid q}1/\phi(q)\le \phi(q^)^{-1}\sum_{d\le Q/q^}1/\phi(d)\ll\phi(q^)^{-1}\log(2Q/q^)\). Split at \(q^\le(\log x)^B\): (SW\(\chi\)) with \(t=x\) gives \(\ll x^2(\log x)^{2B-2H}\) for that part. Above it, partial summation of (LS) against the weight \(\log(2Q/q^)/\phi(q^)\) gives, block by block, \(\sum_{Q'\ \mathrm{dyadic}}(x+Q'^2)x\log x\,\log(2Q/Q')/Q'\ll x^2(\log x)^{2-B}+Qx\log x\), the first term coming from the block just above \((\log x)^B\). Altogether, for every fixed \(A\) and every \(1\le Q\le x\), \[ \boxed{\ D(x,Q)\ \ll_A\ Qx\log x\ +\ x^2(\log x)^{-A}.\ } \tag{BDH\(^\prime\)} \] The second term is the correction. It is not removable by any choice of \(Q\): it is the principal contribution plus the conductors just above the Siegel-Walfisz range, and both are exactly rank 1's kind of obstruction.

Consequence for RANK3_POLYRANGE_TINT_CHECK.md section 3. That section's running sum should read \(F(Q)\ll QN^2L+N^3L^{-A}\), not \(QN^2L\). Carried through its own Abel summation against \(w(q)=\mu(q)^2/\phi(q)^2\), the floor picks up the weight \(W(A)\asymp(\log\log N)^2L^{-2B}\), and the corrected conclusion is \[ S(N,B)\ \ll_A\ N^2L^{1-B}(\log\log N)^2+N^3L^{-A}(\log\log N)^2. \] The boxed claim there, \(S(N,B)=O_\epsilon(N^{2+\epsilon})\), does not follow, and cannot: by (4), \(S(N,B)\ge\sum_{A<q\le R_0}\mu(q)^2\phi(q)^{-3} \sum_{t\le N}(\psi(t,\chi_0)-t)^2\gg L^{-2B}\sum_{t\le N}(\psi(t)-t)^2 -O(NL^{2-2B})\), and section 5 shows \(\sum_{t\le N}(\psi(t)-t)^2\ll N^{3-\delta}\) is a zero-free half-plane. So that box was an RH-strength statement resting on the uncorrected range. The document's numerical section 4, which found \(S(N,1)/N^2\) rising with \(N\), is consistent with the corrected form and was right not to read its own data as confirming the box. RANK3_CROSS_TERM_CANCELLATION.md section 5 and RANK3_MEAN_VALUE_TOOLS.md (E1)-(E2) inherit the same correction: the prime-\(q\) sub-sum is \(O_A(N^3L^{-A})+O(N^2L^{2+o(1)})\), not \(O_\epsilon(N^{2+\epsilon})\), and (E2)'s \(N^{5/2}\) is \(N^{5/2}+N^3L^{-A}\). None of the identities in those documents is affected; only the bounds that consumed (BDH) below its range.

3. The exact decomposition of \(\sum_a^*T_N(q,a)\)

Insert (3) into (2). For squarefree \(q\ge2\), \[ \sum_a^T_N(q,a)\ \le\ \underbrace{\frac{2E(q)}{\phi(q)^3}}{\text{principal}} +\underbrace{\frac2{\phi(q)^3}\sum{\chi\ne\chi_0}\operatorname{cond}(\chi)M_\chi}_{\text{characters}} +\underbrace{\frac{2\rho_2(q)^2N}{\phi(q)}}_{\text{remainder}}. \tag{5} \] This is (2) with (CT0) substituted and nothing discarded. Writing the character part by conductor: for \(\chi\) induced by \(\chi^\bmod q^\), the coefficient sequences of \(K_NL(\chi)\) and \(K_NL(\chi^)\) differ by at most \(\rho_2(q)\) in every entry (the \(2N-1\) entries are partial sums of \(\Lambda(n)\chi(n)\) and their complements, by (1)), so by the triangle inequality in \(\ell^2\), \[ M_\chi\le\big(\sqrt{M_{\chi^}}+\rho_2(q)\sqrt{2N}\big)^2\le2M_{\chi^}+4N\rho_2(q)^2, \tag{6} \] where \(M_{\chi^*}\) is the moment (1) of the primitive character itself. Section 7 checks (6) at \(q\in\{6,10,15\}\); the measured ratio of the two sides is below \(0.46\).

4. The sum over \(2\le q\le R_0\)

The principal part. By (CC-E) in the form \(E(q)\le2T_N+2\rho_2(q)^2N\), \[ \sum_{q=2}^{R_0}\frac{2\mu(q)^2E(q)}{\phi(q)^3} \le4T_N\sum_{\substack{q\ge2\\ q\ \mathrm{squarefree}}}\frac1{\phi(q)^3} +4N\sum_{q\ge2}\frac{\mu(q)^2\rho_2(q)^2}{\phi(q)^3} =4(C_3-1)\,T_N+O(NL^2), \tag{7} \] since \(\sum_{q\ \mathrm{sqfree}}\phi(q)^{-3}=\prod_p(1+(p-1)^{-3})=C_3\) and \(\sum_q\log^2(2q)(1+\log q)^3/q^3\) converges. The \(q=2\) term alone is \(2E(2)/1\le4T_N+O(NL^2)\), and \(C_3-1=1.3009\ldots\) is dominated by it. This is the part of rank 3 that is rank 1 in disguise: the principal character of every modulus carries a copy of \(T_N\), and the weights \(\phi(q)^{-3}\) sum to a constant rather than decaying to zero over the range. No cancellation identity touches it, because (3) already isolates it exactly.

The character part, reorganized by conductor. By (6), \[ \sum_{q=2}^{R_0}\frac{2\mu(q)^2}{\phi(q)^3}\sum_{\chi\ne\chi_0}\operatorname{cond}(\chi)M_\chi \le4\sum_{q=2}^{R_0}\frac{\mu(q)^2}{\phi(q)^3}\sum_{\substack{q^\mid q\\ q^\ge2}}q^* \sum_{\chi^\ \mathrm{prim}\bmod q^}M_{\chi^} +8N\sum_{q=2}^{R_0}\frac{\mu(q)^2\rho_2(q)^2}{\phi(q)^3}\sum_{q^\mid q}q^\phi(q^). \] The last sum is at most \(q^2\), so the second term is \(\ll NL^2\sum_{q\le R_0}\log^2(2q)(1+\log q)^3/q\ll NL^8\). In the first, swap the order: a conductor \(q^*\) is counted once for every squarefree multiple \(q=q^d\le R_0\) with \((d,q^)=1\), so its total weight is \[ \sum_{\substack{q\le R_0\\ q^\mid q}}\frac{\mu(q)^2}{\phi(q)^3} =\frac{\mu(q^)^2}{\phi(q^)^3}\sum_{\substack{d\le R_0/q^\\ (d,q^)=1}}\frac{\mu(d)^2}{\phi(d)^3} \le\frac{C_3}{\phi(q^)^3}. \] Therefore \[ \text{character part}\ \le\ 4C_3\,\Psi+O(NL^8),\qquad \Psi:=\sum_{2\le q^\le R_0}\frac{q^}{\phi(q^)^3}\sum_{\chi^\ \mathrm{prim}\bmod q^}M_{\chi^}. \tag{8} \] This is the reorganization the task asked for: every primitive character appears once, with weight \(q^/\phi(q^)^3\asymp(q^*)^{-2}\) up to \(\log\log\), before anything is bounded. Discarding \(\operatorname{cond}(\chi)\le q\) first, as (CT2) does, would instead put weight \(\asymp1/q\) on every modulus containing the character, and the same conductor would be counted once per multiple; that is the \(\log\log q\) bookkeeping RANK3_COMPOSITE_CROSS_TERM.md section 5 worried about, and it never arises here.

Small conductors, \(2\le q^\le A:=\lfloor L^B\rfloor\). By (1) and (SW\(\chi\)), \(M_{\chi^}\le5N\max_{t\le N}|\psi(t,\chi^)|^2\ll_{B,H}N^3L^{-2H}\) for every primitive \(\chi^\) of conductor \(\le A\), and there are \(\le\phi(q^)\) of them mod \(q^\), so \[ \Psi_{\le A}\ll_{B,H}N^3L^{-2H}\sum_{q^\le A}\frac{q^}{\phi(q^)^2} \ll N^3L^{-2H}\sum_{q^\le A}\frac{(1+\log q^)^2}{q^} \ll N^3L^{-2H}(\log\log N)^3. \tag{9} \]

Large conductors, \(A<q^\le R_0\). Fix \(t\le N\) and put \(g_t(q^)=(q^/\phi(q^))\sum_{\chi^}|\psi(t,\chi^)|^2\ge0\), so that (LS*) reads \(G_t(Q):=\sum_{q^\le Q}g_t(q^)\le(t+Q^2)t\log t\). The weight \(q^/\phi(q^)^3=g\)-weight times \(1/\phi(q^)^2\), and \(1/\phi(q)^2\le W(q):=K(\log\log(q+16))^2/q^2\) with \(K\) absolute (Rosser-Schoenfeld, \(\phi(q)\gg q/\log\log q\)), \(W\) decreasing. Abel summation from \(A\) with \(F(Q)=G_t(Q)-G_t(A)\in[0,G_t(Q)]\): \[ \sum_{A<q^\le R_0}\frac{q^}{\phi(q^)^3}\sum_{\chi^}|\psi(t,\chi^)|^2 \le W(R_0)G_t(R_0)+\sum_{Q=A+1}^{R_0-1}\big(W(Q)-W(Q+1)\big)G_t(Q) \] \[ \ll(\log\log N)^2\Big[\frac{(t+R_0^2)t\log t}{R_0^2} +\sum_{Q>A}\frac{(t+Q^2)t\log t}{Q^3}\Big] \ll(\log\log N)^2\Big[\frac{t^2L}{A^2}+tL^2\Big]. \tag{10} \] The \(t^2L/A^2\) term is the block of conductors just above \(A\), where (LS*) saves nothing because \(A^2\ll t\); the \(tL^2\) term is the sum of \(1/Q\) up to \(R_0\). Applying (10) to the two sums in (1), once summed over \(t\le N\) and once at \(t=N\) with the factor \(2N\): \[ \Psi_{>A}\ll(\log\log N)^2\big[N^3L^{1-2B}+N^2L^2\big]. \tag{11} \]

The remainder. \(2N\sum_q\mu(q)^2\rho_2(q)^2/\phi(q)\ll NL^6\) is (T5).

Assembly. (T2), (5), (7), (8), (9), (11) and (T5) give \[ \sum_{q=2}^{R_0}U_{(q)}\le4(C_3-1)T_N +O_{B,H}\big(N^3L^{-2H}(\log\log N)^3\big) +O\big(N^3L^{1-2B}(\log\log N)^2\big) +O\big(N^2L^2(\log\log N)^2\big)+O(NL^8). \] Given \(A\), take \(B=A+1\), \(H=A\); this is (K). With \(T_N\ll_HN^3L^{-2H}\) from UPPER_BOUND.md section 7, every term but the last two is \(O_A(N^3L^{-A})\). Constants have not been optimized: the factor 4 in (7) is two applications of \((x+y)^2\le2x^2+2y^2\) that an \(\ell^2\) triangle inequality would reduce to \(1+\eta\).

5. The lower bound, and what a power saving would mean

At \(q=2\): \(\phi(2)=1\), \(P_{2,1}=-K_N\), and the coefficient of \(R_{2,1}=F_N(\tfrac12+\beta)+K_N(\beta)\) at \(n\) is \(\Lambda(n)(-1)^n+1=-(\Lambda(n)-1)+2\Lambda(n)\mathbf1_{2\mid n}\). So \(K_NR_{2,1}=-K_N(F_N-K_N)+2K_N\mathcal E\) with \(\mathcal E(\beta)=\sum_{2^k\le N}\log2\cdot\exp1(2^k\beta)\), \(|\mathcal E|\le L\), and \[ T_N(2,1)=\int_{\mathbb T}|K_N|^2|R_{2,1}|^2\ge\big(\sqrt{T_N}-2L\sqrt N\big)^2\ge\tfrac12T_N-4NL^2. \] The arc restriction costs, exactly as in UPPER_BOUND.md (30) with \(\delta_2=Q/(2N)\): \(T_N(2,1)-U_{(2)}\le(N/Q)^2\int_{\mathbb T}|R_{2,1}|^2 \le(N/Q)^2(2d_N+2N)\ll N^2L\). Hence (K'). Section 7 measures \(U_{(2)}/T_N\) at \(N\) up to \(2\times10^4\): it is above \(1\) and decreasing toward it, and (K') holds at every \(N\) tried.

Now suppose \(\sum_{2\le q\le R_0}U_{(q)}\ll N^{3-\delta}\) for one fixed \(0<\delta<1\). By (K'), \(T_N\ll N^{3-\delta}\), so \(\sum_{t\le N}\Delta(t)^2\ll N^{3-\delta}\) with \(\Delta(t)=\psi(t)-t\). For \(\sigma>1-\delta/2\), Cauchy-Schwarz on dyadic blocks gives \(\int_X^{2X}|\Delta(x)|x^{-\sigma-1}dx\le X^{-\sigma-1}\sqrt{X\int_X^{2X}\Delta^2} \ll X^{-\sigma-1}\sqrt{X\cdot X^{3-\delta}}=X^{1-\delta/2-\sigma}\), summable over dyadic \(X\). So \(\int_1^\infty\Delta(x)x^{-s-1}dx\) converges absolutely and is holomorphic in \(\operatorname{Re}s>1-\delta/2\), and by \(-\zeta'/\zeta(s)=s/(s-1)+s\int_1^\infty\Delta(x)x^{-s-1}dx\) (CORRECTED_RH_BRIDGE.md (20)), \(\zeta\) has no zero there. This is the same deduction UPPER_BOUND.md section 1 makes for the total at exponent \(2+\epsilon\) and RESULTS.md section 18 makes for Theorem B, applied at exponent \(3-\delta\). It says nothing about whether such a strip exists; it prices the improvement.

So (K) is the order this route can reach, and the reason is not the cancellation of \(\Sigma_{\rm cross}\), which (3) handles exactly, nor the imprimitive characters, which (8) handles exactly; it is the principal character, present at every modulus with non-decaying total weight \(4(C_3-1)\).

6. Effect on the complete error budget (23)

UPPER_BOUND.md (23): \(E(N)\le32U_Q+8Z_Q+4I_Q+O(N^2L^3)\), with \(U_Q=U_1+\sum_{2\le q\le R_0}U_{(q)}+U_{q>R_0}\). Now \[ U_1\ll_HN^3L^{-2H}\ \text{(section 7 there)},\qquad \sum_{2\le q\le R_0}U_{(q)}\ll_AN^3L^{-A}+N^2L^{2+o(1)}\ \text{(this document)},\qquad U_{q>R_0}\ll N^2L^5\ \text{((27))}. \] So the entire mixed moment is bounded, unconditionally, for every fixed \(A\): \[ \boxed{\ 32U_Q\ll_AN^3L^{-A}+N^2L^5.\ } \tag{12} \] The row "Other moments with \(q\le R_0\): no adequate estimate here" of UPPER_BOUND.md section 8 is, on the \(U\)-side, now filled at rank 1's order, and the composite squarefree moduli are included. What this does to the total:

7. Finite checks

rank3_conductor_sum_probe.py, results in results_rank3_conductor_sum_probe.json, numpy only, a few seconds:

These check identities, an inequality and one measured ratio at small \(N\); they do not test any asymptotic statement.

8. Scope

This document is bookkeeping (orthogonality, induction of characters, Abel summation) around two classical unconditional theorems, (LS*) and Siegel-Walfisz, applied to identities already established on this branch. It corrects one cited range and the three bounds that rested on it, and it establishes an unconditional bound of rank 1's order for the rank-3 mixed moment together with the matching lower bound. It proves no power saving anywhere, claims none, and establishes nothing about the zeros of \(\zeta\) or of any \(L\)-function.