Proxy discrepancy under interval restrictions

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This page reconstructs Lemma 7.3 of the weighted-dilation manuscript at the factor ranges consumed by the extraction of primes 2u-1. It belongs to the elementary estimates of the smooth-successor route. The lemma and its method are OpenAI's; the derivation, the explicit uniformity and the error accounting are Plasma AI's. The E2 examination has a distinct PASS grade, examined on 2026-10-16 UTC, accepting refutation-failed at its stated classical-premise and parameter scopes. The grade retains the original-source reading limits and, for the weighted E3 consumer consequence, its Type II/P hypothesis. The lemma does not involve $2u\pm1$ at all: it concerns ordinary integers in the factor ranges.

Setting

Write $L=\log x$, $T=\log L$, $W=\exp(\sqrt L)$, $V(y)=\prod_{p\le y}(1-1/p)$. Fix $0<b_1<1/10$, $0<\kappa<1/50$ and a finite set $\Gamma$ of exponents $\gamma\in[b_1,1/2-\kappa)$, the left ends of the extraction's bins. The tests are

$$I_\gamma(m)=\mathbf 1_{P^-(m)>x^\gamma}\ (\gamma\in\Gamma),\qquad I_{\rm pr}(m)=\mathbf 1_{m\text{ prime}} .$$

The ambient range is $\mathcal A=[x^{b_1/4},x^{1-b_1/4}]$. Every factor range used by the extraction lies inside it for large $x$: the cofactors of the unbalanced bins lie in $[x^{1/2+\kappa},4x^{1-b_1}]$, the balanced primes in $\mathcal B_\kappa=[x^{1/2-2\kappa},x^{1/2+2\kappa}]$, and a fixed enlargement of every Type II dyad with $x^{b_1/3}\le U\le x^{1-b_1/3}$ is also inside $\mathcal A$.

Fix a precision $S>0$ and put $\Delta=\Delta_L=L^{-S}$. The cells are $C_k=(e^{k\Delta},e^{(k+1)\Delta}]$, $k\in\mathbb Z$; for a cell meeting $\mathcal A$ write $C=(Y,e^\Delta Y]$, so $Y\ge x^{b_1/4}e^{-\Delta}$. There are at most $L^{S+1}+2$ such cells. Put

$$R(C)=\#\{m\in C:P^-(m)>W\},\qquad c_I(C)=\frac{\sum_{m\in C}I(m)}{R(C)},\qquad B_I(m)=c_I(C)\,\mathbf 1_{P^-(m)>W}\quad(m\in C).$$

The counts are full-cell counts. A restriction to an interval $K$ is applied afterwards: the extraction uses $\alpha=(I-B_I)\mathbf 1_K$.

The target is $(\mathrm D_B)$ of the Type II page, the manuscript's (5.4): for every interval $J\subset[1,x^2]$, every Dirichlet character $\chi$ of modulus at most $L^B$, principal and imprimitive ones included, and every real $|t|\le L^B$,

$$\Bigl|\sum_{m\in J}\frac{\alpha_m\chi(m)m^{it}}m\Bigr|\le L^{-B}.$$

Statement

Proposition (proxy discrepancy). Fix $b_1,\kappa,\Gamma$ as above.

1. Uniformly for the cells meeting $\mathcal A$, $R(C)=(e^\Delta-1)YV(W)(1+O(e^{-2T^2}))+O(x^{o(1)})\sim\Delta YV(W)$; in particular $R(C)>0$ for large $x$.
2. Once the denominators are positive as in item 1, $0\le c_I(C)\le1$ for every test and every such cell. Hence $0\le B_I\le1$ and $B_I(m)=0$ unless $P^-(m)>W$. The difference $\alpha=(I-B_I)\mathbf 1_K$ has $|\alpha_m|\le1$ and the same support property. The numerical bound $1$ is independent of $S$; the threshold for positivity may depend on the fixed $S$.
3. For every fixed $B\ge1$ and every fixed $S\ge2B+2$ (in particular for the extraction's choice $S>2B+3$) there is $x_0$ such that for $x\ge x_0$ the sequence $\alpha=(I-B_I)\mathbf 1_K$ satisfies $(\mathrm D_B)$ simultaneously for every test, every interval $K\subset\mathcal A$, every interval $J\subset[1,x^2]$, every character of modulus at most $L^B$ and every $|t|\le L^B$.

The threshold $x_0$ depends on $b_1,\kappa,\Gamma,B$ and $S$. In particular, the coefficient bound and support condition supplied to Type II are independent of $S$, although the proxies themselves and their intermediate error estimates depend on it. Thus $S$ can be fixed after the discrepancy exponent $B$ that premise (P) requires.

Prime sums against characters

The nonprincipal case uses one consequence of the Siegel–Walfisz theorem, the manuscript's Corollary 2.4. It is derived here.

Lemma (harmonic character sums). Fix $c\in(0,1]$, $C_0\ge1$, $B>0$ and $A>0$. For every nonprincipal character $\chi$ of modulus $r\le L^B$, every real $|t|\le L^B$ and every interval $I\subset[\exp(L^c),x^{C_0}]$,

$$\sum_{p\in I}\frac{\chi(p)p^{it}}p\ll_{c,C_0,B,A}L^{-A} .$$

Proof. For $y\ge\exp(L^c)$ the modulus satisfies $r\le(\log y)^{B/c}$. The Siegel–Walfisz theorem (the manuscript's Theorem 2.2) gives, for every fixed $K$, uniformly in $(a,r)=1$, $\pi(y;r,a)=\mathrm{Li}(y)/\varphi(r)+O_K(y(\log y)^{-K})$. A prime dividing $r$ has $\chi(p)=0$, so summing against $\chi(a)$ over the reduced classes cancels the main terms and gives $A_\chi(y)=\sum_{p\le y}\chi(p)\ll_K L^By(\log y)^{-K}$. For $I$ with ends $y_1<y_2$, partial summation against $y^{it-1}$ gives boundary terms $O_K(L^BL^{-cK})$ and an integral at most $(1+|t|)\int_{y_1}^{\infty}|A_\chi(y)|y^{-2}dy\ll_KL^{2B}L^{-c(K-1)}$. Take $cK\ge A+2B+c$. The argument is uniform in both ends and holds for an empty interval, and it uses no primitivity. $\square$

The constants are those of the Siegel–Walfisz theorem and need not be effective.

Proof of the proposition

Item 1. Apply the block sieve of the block-sieve page in its first instance with $r=1$, $J'=C$ and $h=2\lceil T^2\rceil$: the objects are the integers of $C$, $g(p)=1/p$ for $p\le W$, $X=e^\Delta Y-Y$ and $|r(d)|\le1$. This gives the displayed formula with error term at most $D_W=W^{4h+2}=\exp(O(T^2\sqrt L))=x^{o(1)}$. Since $\Delta YV(W)\ge x^{b_1/4}L^{-S}/O(\sqrt L)$, the main term dominates.

Item 2. For $\gamma\in\Gamma$ and large $x$, $x^\gamma\ge x^{b_1}>W$, so $I_\gamma(m)=1$ implies $P^-(m)>W$. A prime $m$ in a cell meeting $\mathcal A$ exceeds $Y\ge x^{b_1/4}/e>W$. So the numerator of $c_I(C)$ counts a subset of the integers counted by $R(C)$.

Item 3, reduction. Fix a test $I$, intervals $K$ and $J$, a character $\chi\bmod r$ with $r\le L^B$, and $|t|\le L^B$. Put $J_0=J\cap K$, an interval inside $\mathcal A$, and

$$\Sigma=\sum_{m\in J_0}(I(m)-B_I(m))\chi(m)m^{it-1}.$$

All bounds below are uniform in these data. Every prime factor of $r$ is at most $L^B<W$, so $(m,r)=1$ whenever $P^-(m)>W$, which holds on the support of $I-B_I$.

Principal characters. Here $\chi(m)=1$ on the support, so $\Sigma=\sum_{m\in J_0}(I(m)-B_I(m))m^{it-1}$. The cells meeting $J_0$ are consecutive; all except at most the first and the last lie inside $J_0$. On a cell $C\subset J_0$ the unweighted sum of $I-B_I$ is $\sum_CI-c_I(C)R(C)=0$, so its contribution is

$$\sum_{m\in C}(I(m)-B_I(m))(m^{it-1}-Y^{it-1}),\qquad |m^{it-1}-Y^{it-1}|\le(1+|t|)\frac{e^\Delta-1}Y .$$

Since $\sum_{m\in C}|I-B_I|\le2R(C)\le2\#(C\cap\mathbb Z)\le2((e^\Delta-1)Y+1)$ and $e^\Delta-1\le2\Delta$, a full cell contributes $O((1+|t|)\Delta(\Delta+x^{-b_1/4}))$ with an absolute constant. Over at most $L^{S+1}+2$ cells the total is $O((1+|t|)L(\Delta+x^{-b_1/4}))=O(L^{B+1-S})$, since $x^{-b_1/4}\le\Delta$ eventually for fixed $S$. Each of the two partial cells contributes at most $\sum_{m\in C}(I+B_I)/m\le2R(C)/Y\le4\Delta+2ex^{-b_1/4}$. With $S\ge2B+2$,

$$|\Sigma|\ll L^{-B-1}+x^{-b_1/4}.$$

Nonprincipal characters: the actual tests. For $I_{\rm pr}$, $\sum_{m\in J_0}I_{\rm pr}(m)\chi(m)m^{it-1}=\sum_{p\in J_0}\chi(p)p^{it-1}$, and $J_0\subset[\exp(L^{1/2}),x]$ for large $x$; the lemma above with $A=B+2$ bounds it by $O(L^{-B-2})$. For $I_\gamma$, every $m\in J_0$ with $P^-(m)>x^\gamma$ factors uniquely as $m=p_1\cdots p_l$ with $x^\gamma<p_1\le\cdots\le p_l$, and $l\le1/\gamma\le1/b_1$ because $m\le x$. Repeated primes are included. Fix $l$ and primes $x^\gamma<p_1\le\cdots\le p_{l-1}$, with product $Q$ ($Q=1$ when $l=1$). The admissible last primes are the primes of

$$I_Q=[p_{l-1},\infty)\cap(x^\gamma,\infty)\cap\{y:Qy\in J_0\},$$

with $p_0=1$; this is one interval inside $(x^\gamma,x]$. By complete multiplicativity,

$$\sum_{m\in J_0}I_\gamma(m)\chi(m)m^{it-1} =\sum_{l\le1/b_1}\ \sum_{x^\gamma<p_1\le\cdots\le p_{l-1}}\chi(Q)Q^{it-1} \sum_{p\in I_Q}\chi(p)p^{it-1}.$$

As $x^\gamma\ge\exp(L^{1/2})$ for large $x$, the lemma with $c=1/2$, $C_0=1$ bounds every inner sum by $O(L^{-A})$ uniformly in $Q$. The outer weights satisfy $\sum1/Q\le(\sum_{x^{b_1}<p\le x}1/p)^{l-1}\le(\log(1/b_1)+1)^{l-1}$ by Mertens' estimate. Taking $A=B+2$, the actual test contributes $O_{b_1}(L^{-B-2})$.

Nonprincipal characters: the proxy. Fix a cell $C=(Y,e^\Delta Y]$ meeting $J_0$ and put $J'=C\cap J_0$. For $y\in J'$ let $J'_y=J'\cap(-\infty,y]$ and

$$F(y)=\sum_{m\in J'_y}\mathbf 1_{P^-(m)>W}\chi(m) =\sum_{(a,r)=1}\chi(a)\,N(J'_y;a),\qquad N(J'';a)=\#\{m\in J'':m\equiv a\ (r),\ P^-(m)>W\}.$$

The first instance of the block sieve, for the class $a$ and the interval $J'_y$, gives

$$N(J'_y;a)=\frac{|J'_y|\,V(W)}{\varphi(r)}\bigl(1+\theta_ac_1e^{-h}\bigr) +\theta'_aD_W,\qquad|\theta_a|,|\theta'_a|\le1,$$

because $X\prod_{p\le W,,p\nmid r}(1-1/p)=(|J'y|/r)V(W)\prod{p\mid r}(1-1/p)^{-1}=|J'_y|V(W)/\varphi(r)$. This holds for every subinterval, however short, as an absolute-error statement; no relative asymptotic is claimed on short intervals. The main term does not depend on $a$, and $\chi$ is nonprincipal, so it cancels:

$$|F(y)|\le c_1e^{-h}|J'|V(W)+\varphi(r)D_W .$$

Partial summation on $J'\subset(Y,e^\Delta Y]$ against $m^{it-1}$ gives

$$\Bigl|\sum_{m\in J'}\mathbf 1_{P^-(m)>W}\chi(m)m^{it-1}\Bigr| \le\sup_{y\in J'}|F(y)|\,\frac{1+(1+|t|)(e^\Delta-1)}Y \le\frac{2}{Y}\bigl(c_1e^{-h}|J'|V(W)+L^BD_W\bigr)$$

once $(1+|t|)(e^\Delta-1)\le1$, which holds for $S>B$ and large $x$. Multiply by $c_I(C)\in[0,1]$ and sum over the at most $L^{S+1}+2$ cells, using $|J'|\le(e^\Delta-1)Y+1$:

$$\Bigl|\sum_{m\in J_0}B_I(m)\chi(m)m^{it-1}\Bigr| \ll e^{-2T^2}LV(W)+L^{S+1+B}D_Wx^{-b_1/4}\ll L^{-B-2}.$$

The first term is smaller than every fixed power of $L$ because $e^{-2T^2}=L^{-2T}$; the second is $x^{-b_1/4+o(1)}$ times a fixed power of $L$.

Conclusion. In all cases $|\Sigma|\le L^{-B}$ for $x\ge x_0$, with $x_0$ depending only on $b_1,\kappa,\Gamma,B,S$ and the Siegel–Walfisz constants. Since $J_0=J\cap K$, this is $(\mathrm D_B)$ for $\alpha=(I-B_I)\mathbf 1_K$. $\square$

Uniformity checks

• Restrictions. $K$ may cut cells anywhere; at most two cut cells meet $J\cap K$, and they enter only through absolute bounds. The proxy is not rescaled on a cut cell. The proof covers every interval $K\subset\mathcal A$, which contains every restriction the extraction makes.
• Characters. Principal characters of every modulus up to $L^B$ act as the constant $1$ on the support. Every nonprincipal character, primitive or not, is handled by the same two arguments: the lemma above needs no primitivity, and the residue-class cancellation uses only $\sum_{(a,r)=1}\chi(a)=0$.
• The order of $B$ and $S$. The coefficient bound $|\alpha_m|\le1$, the support condition and the admissible $K$ are fixed before $S$; the losses $L^{S+1}$ from the number of cells meet savings $e^{-2T^2}$ and $x^{-b_1/4+o(1)}$ that beat every fixed power of $L$, and the principal case needs only $S\ge2B+2$. No relation between $S$ and $B$ feeds back into the Type II coefficient bound.
• The sign of the shift. No step involves $2u\pm1$.