Marked Type II estimates and rough factors

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The construction of Lessons 18 and 19 counts integers \(d\) whose predecessor \(d-1\) has a prescribed shape, and then removes the composite \(d\). Both steps require bilinear sums \(\sum_{m,n}\alpha_m\beta_n\,F(mn-1)\mathcal W(mn-1)\) to be small, where \(\mathcal W\) is a weight that marks prime factors of \(mn-1\) from fixed groups of primes. A theorem of OpenAI's companion paper on the Poisson–Dirichlet law for prime predecessors gives this when \(\alpha_m\) is the indicator of primes minus its average over \(W\)-rough numbers. Its proof uses \(\alpha\) only through cancellation against Dirichlet characters of small modulus twisted by \(m^{it}\). Both facts are theorems of The marked Type II estimate, the first lesson of the course Marked Type II estimates for prime predecessors, which proves them. This lesson states them in the form used here and proves the corresponding cancellation for the coefficient that replaces the indicator of \(x^\gamma\)-rough numbers by its density \(D_\gamma(\log m/L)/L\) of A block sieve and rough-number densities.

Prerequisites. A block sieve and rough-number densities, (1.1)–(1.4), Lemma 1.1 and Lemma 2.1; The Siegel–Walfisz theorem, Theorem 2.1; Counting primes by elementary means: Chebyshev and Mertens, Theorems 3.1 and 4.2; The marked Type II estimate, Theorems 2.2 and 2.3; A mean value theorem and logarithmic phases at large height, Theorem 2.1; Long prime polynomials, Theorem 1.1.

1. Marked weights

Keep \(L=\log x\), \(W=\exp(L^{0.24})\) and \(V(y)=\prod_{p\le y}(1-1/p)\) from Lesson 16. Dirichlet characters, principal ones included, are extended by zero on nonunits. Fix an integer \(K\ge1\) and numbers \(0.1<a_1<\cdots<a_K<0.2\) before letting \(x\) grow, and define the groups \[ \begin{gathered} \mathcal P_i=\{p\ \text{prime}:L^{a_i}\le\log p\le2L^{a_i}\}\\ (1\le i\le K),\quad(1.1) \end{gathered} \] which are disjoint for large \(x\). For \(h\ge1\) and \(0<q<1\) put \[ \begin{gathered} V_i=\sum_{p\in\mathcal P_i}\frac1p,\\ \omega_i(h)=\#\{p\in\mathcal P_i:p\mid h\},\\ \omega=\sum_i\omega_i,\\ \mathcal W(h)=q^{\omega(h)-K}\prod_{i=1}^K\frac{\omega_i(h)}{V_i}.\quad(1.2) \end{gathered} \] By Mertens' estimate (Theorem 3.1 of Counting primes by elementary means), \(V_i=\log2+o(1)\). The weight \(\mathcal W\) is nonnegative, vanishes unless every group supplies a prime divisor, and is bounded in terms of \(K\) and \(q\), since \(jq^{j-1}\) is bounded. A function \(F\) on positive integers is group-invariant if \(F(ph)=F(h)\) for every group prime \(p\) and every \(h\); equivalently, \(F(h)\) depends only on \(h\) with all its group prime powers removed.

2. The marked Type II estimate and long prime polynomials

N5 (one-sided marked Type II estimate [PD, Theorem 3.1]; Theorem 2.3 of The marked Type II estimate). Fix \(\delta,C,D_*>0\) and \(q\in(0,1)\). Let \(H_m,H_n\ge x^\delta\) with \(X=H_mH_n\asymp x\). Let \(F\) be group-invariant with \(|F|\le1\). Let \(\alpha_m\) be supported on an interval in \([H_m,2H_m]\), where \[ \begin{gathered} \begin{aligned} \alpha_m&=m^{iv}\Bigl(\mathbf 1_{m\ \text{prime}}\\ &\quad -\frac{\mathbf 1_{P^-(m)>W}}{V(W)\log m}\Bigr), \end{aligned}\\ |v|\le L^C, \end{gathered} \] and let \(\beta_n\) be supported on \(W\)-rough integers in \([H_n,2H_n]\) with \(|\beta_n|\le L^C\). If \(K\) is large enough in terms of \(\delta,C,D_*,q\), then \[ \begin{aligned} &\Bigl|\sum_{m,n}\alpha_m\beta_n\,F(mn-1)\,\mathcal W(mn-1)\Bigr|\\ &\quad \le C'\,X\,L^{-D_*}.\quad(2.1) \end{aligned} \] The lower bound for \(K\) does not depend on the \(a_i\); the constant \(C'\) and the threshold for \(x\) may.

N6 (coefficient criterion [OpenAI-029, Lemma 10.2], extracted from the proof of N5 in [PD, Sections 3–5]; Theorem 2.2 of The marked Type II estimate). In N5, replace \(\alpha\) by any sequence supported on \(W\)-rough integers of an interval \(I\subseteq[H_m,2H_m]\), keep \(\beta\) as there, and assume \(|\alpha_m|\le L^C\). Suppose that for all fixed \(A_0,B,A>0\), every Dirichlet character \(\chi\) of modulus \(k\le L^{A_0}\) and every \(|t|\le2XL^B\), \[ \begin{aligned} &\Bigl|\frac1{H_m}\sum_m\alpha_m\chi(m)m^{it}\Bigr|\\ &\quad \le C_A\,L^{-A},\quad(2.2) \end{aligned} \] uniformly in the interval, the character and \(t\). Then (2.1) holds for \(K\) large enough in terms of the fixed data, with the same uniformity.

N7 (long prime polynomials and logarithmic phases [PD, Lemmas 2.3 and 2.4], proved in [PD, Appendix, Lemmas 9.1 and 9.5]; also [PP, Lemmas 3.1 and 3.3]; Theorem 1.1 of Long prime polynomials and Theorem 2.1 of A mean value theorem and logarithmic phases at large height).

  1. Fix \(0<\tau<\theta<1\) and \(C,A>0\). There is \(B_0\) such that for \(x^\tau/2\le N\le x^\theta\), every character \(\chi\) of modulus \(k\le L^C\), every interval \(J\subseteq[N,2N]\) and \(L^{B_0}\le|u|\le x^2\), \[ \Bigl|\sum_{p\in J}\chi(p)\,p^{-1+iu}\Bigr|\le C_{\tau,\theta,C,A}\,L^{-A}. \]
  2. Fix \(C>0\). There is an absolute \(C_3>0\) such that for large \(x\), \(\exp(L/(\log L)^2)\le N\le2x^5\), \(k\le L^C\), \(\exp(L/(2(\log L)^2))\le|u|\le4x^3\), every residue \(a\) modulo \(k\) and every interval \(J\subseteq[N,2N]\), \[ \begin{aligned} &\Bigl|\sum_{\substack{n\in J\\ n\equiv a\ (k)}}n^{iu}\Bigr|\\ &\quad \le\frac Nk\exp\Bigl(-\frac L{(\log L)^{C_3}}\Bigr). \end{aligned} \]

The course Marked Type II estimates for prime predecessors proves all three, in the following form. Its Theorem 2.2 is N6 without the restriction to an interval: the coefficient may be any sequence on the \(W\)-rough integers of \([H_m,2H_m]\). In Theorems 2.2 and 2.3 the number of groups needed depends only on \(C\) and \(D_*\). Theorem 1.1 of Long prime polynomials holds for \(x^\tau/2\le N\le x^{1/\tau}\), which contains the range of N7(1), with \(B_0=A+8\). Theorem 2.1 of A mean value theorem and logarithmic phases at large height is N7(2), with \(C_3=10\). N5 and N6 rest on a determinant estimate with marks on one side and on a major-arc analysis; N7 rests on Vinogradov's mean-value method and a zero-free strip at large height.

3. Characters on rough numbers

Lemma 3.1 (rough counts [PD, Lemma 2.5]). Fix \(\delta,C,A_0,A>0\). For \(x^\delta\le H\le x^C\), every interval \(I\subseteq[H,2H]\) and every character \(\chi\) of modulus \(k\le L^{A_0}\), \[ \begin{aligned} &\sum_{\substack{n\in I\\ P^-(n)>W}}\chi(n)\\ &\quad =\mathbf 1_{\chi\ \text{principal}}\,V(W)\,|I|\\ &\qquad +O(H\,L^{-A}).\quad(3.1) \end{aligned} \]

Proof. For large \(x\), every prime factor of \(k\) is below \(W\), so a \(W\)-rough \(n\) is a unit modulo \(k\) and the principal character is \(1\) on it. Fix a unit residue \(b\) modulo \(k\), and apply Lemma 2.1 of Lesson 16 to \(\{n\in I:n\equiv b\ (k)\}\) with the primes \(p\le W\), \(p\nmid k\), and the conditions \(p\mid n\). By the Chinese remainder theorem, \(A(d)=|I|/(kd)+O(1)\), so \(X'=|I|/k\), \(g(d)=1/d\), \(|E(d)|\le1\), and (2.1) of Lesson 16 holds with absolute constants. Take the sieve depth \(2\lceil(A+A_0)\log L\rceil\), so that its relative error is \(O(L^{-A-A_0})\) and the remainder sum is at most \(W^{O(\log L)}=x^{o(1)}\). Since \(\prod_{p\le W,\,p\nmid k}(1-1/p)=V(W)k/\varphi(k)\), the count in each unit class is \(V(W)|I|/\varphi(k)+O(HL^{-A-A_0})\). Multiply by \(\chi(b)\) and sum over the \(\varphi(k)\le L^{A_0}\) classes. \(\square\)

4. The rough proxy at high frequencies

Lemma 4.1 (the proxy bound [PD, proof of Lemma 5.2]). Fix \(\delta,A_0>0\), let \(x^\delta\le M\le x\), and put \(T_*=\exp(L/(\log L)^2)\). There are \(C_0,c_0>0\), depending on \(\delta,A_0\), such that for every fixed \(A_4\), every interval \(I'\subseteq[M,2M]\), every character \(\chi\) of modulus \(k\le L^{A_0}\) and \(1\le|u|<x^2\), \[ \begin{aligned} &\Bigl|\frac1{M\,V(W)}\sum_{\substack{m\in I'\\ P^-(m)>W}}\frac{\chi(m)\,m^{iu}}{\log m}\Bigr|\\ &\quad\le C\,L^{-A_4}+C\,L^{C_0}\Delta(u),\quad(4.1) \end{aligned} \] where \(C_3\) is the constant of N7 and \[ \Delta(u)=\begin{cases}|u|^{-1}+x^{-c_0},&|u|\le T_*,\\ e^{-L/(\log L)^{C_3}},&T_*\le|u|<x^2 .\end{cases} \]

Proof. Truncation. Let \(S_W=\sum_{p\le W}1/p\), which is \(0.24\log L+O(1)\) by Mertens, and for an integer \(r\) put \(T_r(m)=\sum\mu(d)\) over the squarefree \(d\mid m\) composed of primes \(p\le W\) with at most \(r\) prime factors. If \(m\) has \(s\) prime factors \(p\le W\), then \(T_r(m)=\sum_{a\le r}(-1)^a\binom sa\), which differs from \(\mathbf 1_{s=0}\) by at most \(\binom s{r+1}\) (Exercise 4.1 of Lesson 16). Summing \(\binom s{r+1}\), the number of divisors \(d\) with \(r+1\) such primes, over \(m\in I'\), \[ \begin{aligned} &\sum_{m\in I'}\bigl|\mathbf 1_{P^-(m)>W}-T_r(m)\bigr|\\ &\quad \le\sum_{\omega(d)=r+1}\Bigl(\frac{2M}d+1\Bigr)\\ &\quad \le2M\frac{S_W^{r+1}}{(r+1)!}+W^{r+1}.\quad(4.2) \end{aligned} \] With \(r=\lceil c\log L\rceil\) and \(c\) large, the first term is \(O(ML^{-A_4-1})\), and the second is \(\exp(O(L^{0.24}\log L))=x^{o(1)}\). Moreover \(\sum_d1/d\le\prod_{p\le W}(1+1/p)\le CL^{0.24}\) over all \(d\) used, and \(1/V(W)\le CL^{0.24}\).

One divisor. For a used \(d\) prime to \(k\) (the others give \(\chi(m)=0\)), write \(m=dz\) and split \(z\) into residues \(a\) modulo \(k\); the range of \(z\) is an interval of scale \(N'=M/d\ge x^{\delta}/W^{r}\ge x^{\delta/2}\). For \(|u|\le T_*\), comparison with the integral gives \[ \begin{gathered} \begin{aligned} \sum_{\substack{z\in J\\ z\equiv a\ (k)}}z^{iu}&=\frac1k\int_Jy^{iu}\,dy+O(1+|u|), \end{aligned}\\ \Bigl|\int_Jy^{iu}\,dy\Bigr|\le\frac{4N'}{|u|}, \end{gathered} \] since \(y^{iu}\) has total variation \(O(|u|)\) on \(J\) and an antiderivative \(y^{1+iu}/(1+iu)\). For \(T_*\le|u|<x^2\), N7(2) applies, because \(N'\ge x^{\delta/2}\ge\exp(L/(\log L)^2)\) and \(|u|\ge T_*\). The factor \(d^{iu}\chi(d)\) has absolute value at most \(1\), and partial summation handles \(1/\log m\), which has bounded variation relative to its size on the dyad.

Assembling. Summing over the residues and over the \(d\le W^r\) with the weight bounds above, the error terms \(O(1+|u|)\) contribute \(O(kW^rT_*/M)=O(x^{-c_0})\) after division by \(M\), and the main terms contribute at most \(CL^{C_0}|u|^{-1}\), respectively \(CL^{C_0}\exp(-L/(\log L)^{C_3})\), with \(C_0\) accounting for \(L^{0.24}\cdot L^{0.24}\cdot L^{A_0}\) and the partial summation. The truncation error (4.2) gives the term \(L^{-A_4}\). \(\square\)

5. A rough-factor Type II estimate

Let \(D_\gamma\) be the densities (1.3) of Lesson 16.

Proposition 5.1 (rough-factor Type II estimate [OpenAI-029, Proposition 10.3]). Keep the marked data and the hypotheses on \(F,\beta,H_m,H_n\) of N5. Fix \(0<\gamma<w_-<w_+<1\) with \([\log H_m/L,\log(2H_m)/L]\subseteq[w_-,w_+]\). On an interval \(I\subseteq[H_m,2H_m]\) put \[ \begin{gathered} \begin{aligned} \alpha_m&=m^{iv}\Bigl(\mathbf 1_{P^-(m)>x^\gamma}\\ &\quad -\frac{D_\gamma(\log m/L)}{L\,V(W)}\mathbf 1_{P^-(m)>W}\Bigr), \end{aligned}\\ |v|\le L^C,\quad(5.1) \end{gathered} \] and \(\alpha_m=0\) outside \(I\). Then (2.1) holds for every fixed \(D_*\) once \(K\) is large in terms of the fixed data, with a lower bound for \(K\) independent of the \(a_i\).

Proof. We verify (2.2) and apply N6. Put \(M=H_m\), \(R_\gamma(m)=\mathbf 1_{P^-(m)>x^\gamma}\), and \(J=\lfloor w_+/\gamma\rfloor\); an \(x^\gamma\)-rough \(m\le2M\) has at most \(J\) prime factors.

Prime products. Let \(T_\gamma(m)\) be the sum over \(j\le J\) of \(1/j!\) times the number of ordered \(j\)-tuples \((p_1,\dots,p_j)\) of primes \(p_i>x^\gamma\) with \(p_1\cdots p_j=m\). For squarefree \(x^\gamma\)-rough \(m\) this is \(1\); for an \(x^\gamma\)-rough \(m\) with multiplicities \(e_p\) it is \(1/\prod e_p!\); and it is \(0\) if \(m\) is not \(x^\gamma\)-rough. So \(R_\gamma-T_\gamma\) is supported on \(m\) divisible by \(p^2\) for some \(p>x^\gamma\), and on every subinterval of the dyad \[ \begin{aligned} &\sum_{M\le m\le2M}|R_\gamma(m)-T_\gamma(m)|\\ &\quad \le\sum_{p>x^\gamma}\frac{2M}{p^2}\le C\,M\,x^{-\gamma},\quad(5.2) \end{aligned} \] also after any twist of absolute value at most \(1\).

Low frequencies: the rough count. For fixed \(A_0,A_1\), every \(k\le L^{A_0}\), every \(\chi\) modulo \(k\) and every interval \(I'\subseteq[M,2M]\), \[ \begin{aligned} &\sum_{m\in I'}R_\gamma(m)\chi(m)\\ &\quad =\mathbf 1_{\chi\ \text{principal}}\frac1L\int_{I'}D_\gamma\Bigl(\frac{\log y}L\Bigr)\\ &\qquad dy+O(ML^{-A_1}).\quad(5.3) \end{aligned} \] To see this, use \(T_\gamma\) and (5.2). Split each prime variable of the \(j\)-prime term into dyads \([P_i,2P_i)\), with \(x^\gamma/2\le P_i\le2M\); there are \(O(L^j)\) boxes. On a dyad of scale \(P\), \(\log P\asymp L\), and the Siegel–Walfisz theorem (Theorem 2.1 of The Siegel–Walfisz theorem), converted to primes by partial summation and to characters by orthogonality over the at most \(L^{A_0}\) classes, gives for every subinterval \(J'\subseteq[P,2P]\) \[ \begin{aligned} \sum_{p\in J'}\chi(p)&=\mathbf 1_{\chi\ \text{principal}}\int_{J'}\frac{dy}{\log y}\\ &\quad +O(PL^{-A_2}), \end{aligned} \] with \(A_2\) arbitrary; the primes involved exceed \(k\). In a box, replace the prime measures one at a time: with the other variables fixed, the product condition leaves an interval for the remaining one, the error is \(O(P_iL^{-A_2})\), and the other variables have total mass \(O(P_h)\); so each box has error \(O(ML^{-A_2})\), and \(A_2\) large absorbs the \(O(L^j)\) boxes. For nonprincipal \(\chi\) the main terms vanish. For principal \(\chi\) the main term of the \(j\)-prime part is \(\frac1{j!}\int_{y_i>x^\gamma,\ \prod y_i\in I'}\prod dy_i/\log y_i\), and Lemma 1.1 of Lesson 16 sums these to the main term of (5.3).

Low frequencies: cancellation. Let \(u=t+v\) and fix \(B_2>C+2\). If \(|t|\le L^{B_2}\), then \(|u|\le2L^{B_2}\). Partial summation of (5.3) against \(y^{iu}\), whose total variation on the dyad is \(O(1+|u|)\), gives \(\mathbf 1_{\chi\ \text{principal}}\frac1L\int_ID_\gamma(\log y/L)y^{iu}dy\)\(+O(ML^{-A_1}(1+|u|))\). Partial summation of (3.1) against \(D_\gamma(\log y/L)y^{iu}/(LV(W))\) gives the same main term with an error \(O(ML^{-A}L^{C'})\): \(D_\gamma\) is Lipschitz on \([w_-,w_+]\) (Lemma 1.1 of Lesson 16), and \(LV(W)\asymp L^{0.76}\) by Mertens' product theorem. The two main terms cancel in \(\alpha\), so (2.2) holds for \(|t|\le L^{B_2}\) once \(A_1\) and \(A\) are chosen after \(B_2\).

High frequencies. Choose \(0<\tau<\gamma\) and \(w_+<\theta<1\). Let \(L^{B_2}\le|t|\le2XL^B\); then \(|u|\asymp|t|\) and \(L^{B_0}\le|u|<x^2\), where \(B_0\) is taken from N7(1) for the accuracy \(A_3\). Every prime dyad of the convolution lies in \([x^\tau/2,x^\theta]\), and N7(1) with partial summation gives \(|\sum_{p\in J'}\chi(p)p^{iu}|\le CPL^{-A_3}\) on every subinterval. In each box fix all primes but one, bound that one slot by this estimate and the others trivially: the box contributes \(O(ML^{-A_3})\), and with (5.2) the prime-product part of \(\alpha\) has the saving \(L^{-A}\) for \(A_3\) large. For the proxy part write \[ \begin{gathered} \frac{D_\gamma(w)}{L\,V(W)}=\frac{w\,D_\gamma(w)}{V(W)\log m}\\ (w=\log m/L); \end{gathered} \] the factor \(wD_\gamma(w)\) is bounded with bounded variation on \([w_-,w_+]\), so Lemma 4.1 and one partial summation bound this part by \(C(L^{-A_4}+L^{C_0}|u|^{-1}\)\(+L^{C_0}x^{-c_0}+L^{C_0}e^{-L/(\log L)^{C_3}})\), which is \(O(L^{-A})\) when \(B_2>A+C_0+2\) and \(A_4\ge A\). Choosing first \(A\), then \(A_3,A_4\) and \(B_0\), and finally \(B_2>\max\{C+2,B_0+2,A+C_0+2\}\) covers all \(|t|\le2XL^B\). The conjugated coefficients satisfy the same bounds, with \(\overline\chi\) and \(-t\).

Support and size. For large \(x\), \(x^\gamma>W\), so both parts of (5.1) are supported on \(W\)-rough integers, and \(|\alpha_m|\le1+\sup D_\gamma/(LV(W))\le2\). With these hypotheses verified, N6 gives (2.1). \(\square\)

The point of Proposition 5.1 is that \(\mathbf 1_{P^-(m)>x^\gamma}\), unlike the prime indicator of N5, can be computed by a sieve but has no cancellation of its own; the subtraction of \(D_\gamma(\log m/L)/(LV(W))\) times the \(W\)-rough indicator removes exactly its main terms, leaving a coefficient that the marked bilinear machinery can absorb.

6. Exercises

6.1. Show that \(jq^{j-1}\le\max(1,\,1/(e\,q\log(1/q)))\) for \(j\ge1\) and \(0<q<1\), and deduce a bound for \(\mathcal W\) in terms of \(K\) and \(q\).

6.2. Show that a function \(F\) on positive integers satisfies \(F(ph)=F(h)\) for all group primes \(p\) if and only if \(F(h)=F(h/h_{\mathcal P})\), where \(h_{\mathcal P}\) is the largest divisor of \(h\) composed of group primes.

6.3. In (5.2), show that \(\sum_{p>y}p^{-2}\le2/y\) for \(y\ge2\).

6.4. In Lemma 4.1, check that \(S_W^{r+1}/(r+1)!\le L^{-A}\) when \(r=\lceil c\log L\rceil\) with \(c\) large, using \(n!\ge(n/e)^n\).

6.5. For the coefficient of N5 with \(v=0\), \(\alpha_m=\mathbf 1_{m\ \text{prime}}\)\(-\mathbf 1_{P^-(m)>W}/(V(W)\log m)\), show that (2.2) holds for \(|t|\le L^{B'}\) and every fixed \(B'\), using the Siegel–Walfisz theorem and Lemma 3.1.

7. Solutions

6.1. The maximum of \(jq^{j-1}\) over real \(j\ge1\) is at \(j=1/\log(1/q)\) when this exceeds \(1\), with value \(1/(eq\log(1/q))\). Then \(\mathcal W(h)\le q^{-K}\prod_i\omega_i(h)q^{\omega_i(h)}/V_i\)\(\le\bigl(\max_j jq^{j-1}\bigr)^K\prod_iV_i^{-1}\), bounded since \(V_i\to\log2\).

6.2. If \(F(ph)=F(h)\) for all group primes, remove the group prime factors of \(h\) one at a time. Conversely \(ph\) and \(h\) have the same quotient by their group parts.

6.3. \(\sum_{p>y}p^{-2}\le\sum_{n>y}n^{-2}\)\(\le\int_{y-1}^\infty t^{-2}\,dt=1/(y-1)\)\(\le2/y\).

6.4. With \(S_W\le\log L\) and \(n=r+1\ge c\log L\), \(S_W^n/n!\le(eS_W/n)^n\le(e/c)^{c\log L}\)\(=L^{-c\log(c/e)}\), which is below \(L^{-A}\) for \(c\) large.

6.5. By Siegel–Walfisz, \(\sum_{p\in I'}\chi(p)\)\(=\mathbf 1_{\chi\ \text{principal}}\int_{I'}dy/\log y\)\(+O(H_mL^{-A})\) on every subinterval, and by Lemma 3.1, \(\sum_{m\in I',\,P^-(m)>W}\chi(m)\)\(=\mathbf 1_{\chi\ \text{principal}}V(W)|I'|\)\(+O(H_mL^{-A})\). Partial summation against \(y^{it}\) (variation \(O(1+|t|)\le O(L^{B'})\)) and against \(y^{it}/(V(W)\log y)\) turns both into \(\mathbf 1_{\chi\ \text{principal}}\int_Iy^{it}dy/\log y\)\(+O(H_mL^{-A+B'+1})\); the main terms cancel, and \(A\) is arbitrary.

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