Weights, mass and distribution in a progression

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

This lesson sets up the construction that proves Proposition 5.1 of Artin's conjecture and the index of a prime. Integers \(d=crQ+1\) in a fixed progression, with \(Q>x^{0.9}\) prime and \(r\) composed of primes from the groups of Marked Type II estimates and rough factors, receive nonnegative weights \(w(d)\) built from the marked weight \(\mathcal W\). Three facts about these weights are proved here: the marked Type II estimate holds inside the progression; the total weight \(X_0\) is of order \(xJ_0/L\), where \(J_0\) is a harmonic sum over the group integers \(r\); and the family is well distributed in the sense of the Bombieri–Vinogradov theorem, which is what the sieve of Lesson 19 needs. Lesson 19 then shows that a positive proportion of the weight sits on primes.

Prerequisites. Artin's conjecture and the index of a prime, (4.1) and Proposition 5.1; A block sieve and rough-number densities, (1.1)–(1.3); Marked Type II estimates and rough factors, Section 1, N6 and Proposition 5.1; The Bombieri–Vinogradov theorem, Theorem 3.1 and Corollary 4.1; The Siegel–Walfisz theorem, Theorem 2.1; Counting primes by elementary means: Chebyshev and Mertens, Theorems 3.1 and 4.2; Schwartz functions and Fourier inversion, Section F4.

1. The weights

Fix \(M,c,u\) as in (4.1) of Lesson 1: \[ \begin{gathered} 8\mid M,\quad c\in\{2,4\},\quad(u,M)=1,\\ c\mid u-1,\quad\Bigl(\frac{u-1}c,\frac Mc\Bigr)=1 .\quad(1.1) \end{gathered} \] Use \(L\), \(W\), \(V\) of Lesson 16 and the groups \(\mathcal P_i\) and the marked weight \(\mathcal W\) of Lesson 17 with \(q=\frac12\); the integer \(K\) and the exponents \(0.1<a_1<\cdots<a_K<0.2\) are fixed before \(x\) grows and chosen at the end of Lesson 19. For large \(x\) the group primes lie in \((\exp(L^{0.1}),\exp(L^{0.3}))\), so they do not divide \(M\). A group integer is a positive integer all of whose prime factors are group primes (including \(1\)), and \(h_{\mathcal P}\) is the largest group-integer divisor of \(h\).

Fix a smooth \(\Psi\) supported in \((1,2)\) with \(0\le\Psi\le1\) and \(A_\Psi=\int_1^2\Psi>0\), and put \[ \begin{aligned} &F(h)=\mathbf 1_{\{h/h_{\mathcal P}=cQ,\ Q>x^{0.9}\ \text{prime}\}},\\ &w(d)=\mathbf 1\{d\equiv u\ (M)\}\,\Psi(d/x)\\ &\qquad\times F(d-1)\,\mathcal W(d-1)\\ &\qquad(d\ge2),\quad(1.2) \end{aligned} \] and \(w(1)=0\). Since \((ph)_{\mathcal P}=p\,h_{\mathcal P}\) for every group prime \(p\), \(F\) is group-invariant: \(F(ph)=F(h)\).

Lemma 1.1 (support). If \(w(d)\ne0\), then \(d=crQ+1\) with \(r=(d-1)_{\mathcal P}\) a group integer and \(Q>x^{0.9}\) prime, uniquely, and \[ \begin{gathered} r\le x^{0.11},\\ \mathcal W(d-1)=\mathcal W(r),\\ 0\le w(d)\le B_K ,\quad(1.3) \end{gathered} \] with \(B_K\) depending only on \(K\).

Proof. Neither \(c\) nor \(Q\) has a group prime factor (group primes are odd and smaller than \(x^{0.9}\)), so \(r=(d-1)_{\mathcal P}\), the group primes of \(d-1\) are those of \(r\), and \(r\le2x/x^{0.9}\). Finally \(\omega_i(h)2^{1-\omega_i(h)}\le1\), so \(\mathcal W\le\prod_iV_i^{-1}\), which is bounded since \(V_i\to\log2\). \(\square\)

The total weight is \(X_0=\sum_dw(d)\). The aim of Lessons 18–19 is \(\sum_{d\ \text{prime}}w(d)\gg X_0/L\), together with \(X_0\gg x/L\); with \(w\le B_K\) this gives \(\gg x/L^2\) primes.

2. The marked estimate in the progression

For fixed \(\gamma>0\) and integers \(m\), put \[ \begin{gathered} R_\gamma(m)=\mathbf 1\{P^-(m)>x^\gamma\},\\ \begin{aligned} B_\gamma(m)&=\frac{D_\gamma(\log m/L)}{L\,V(W)}\\ &\quad \,\mathbf 1\{P^-(m)>W\}, \end{aligned}\quad(2.1) \end{gathered} \] with the densities \(D_\gamma\) of Lesson 16.

Lemma 2.1 (Type II in the progression [OpenAI-029, Lemma 12.1]). Fix \(\delta>0\) and \(0<\gamma<s_-<s_+<1\). Let \(H_m,H_n\ge x^\delta\) with \(H_mH_n\asymp x\), let \(I\subseteq[H_m,2H_m]\) be an interval with \([\log H_m/L,\log(2H_m)/L]\subseteq[s_-,s_+]\), and let \(|b_n|\le1\) be supported on \(W\)-rough integers in \([H_n,2H_n]\). There is \(K_0\), independent of the \(a_i\), such that for \(K\ge K_0\) \[ \begin{aligned} &\Bigl|\sum_{m\in I,\,n}\bigl(R_\gamma(m)-B_\gamma(m)\bigr)\,b_n\,w(mn)\Bigr|\\ &\quad \le C\,x\,L^{-6},\quad(2.2) \end{aligned} \] uniformly in the dyads, the interval and the coefficients.

Proof. Since \((u,M)=1\), orthogonality of the characters \(\lambda\) modulo \(M\) gives \(\mathbf 1\{mn\equiv u\ (M)\}\)\(=\varphi(M)^{-1}\sum_\lambda\overline{\lambda(u)}\lambda(m)\lambda(n)\). Let \(\widehat\psi(v)=\frac1{2\pi}\int_{\mathbb R}\Psi(e^s)e^{-ivs}\,ds\), a Schwartz function; Fourier inversion in \(\log y\) (Section F4 of Schwartz functions and Fourier inversion) gives \(\Psi(mn/x)=\int_{\mathbb R}\widehat\psi(v)x^{-iv}m^{iv}n^{iv}\,dv\). For \(|v|\le L\) and each \(\lambda\), apply N6 of Lesson 17 to the group-invariant \(F\) of (1.2), with \(C=2\), \(D_*=6\), \(\alpha_m=\mathbf 1_I(m)m^{iv}\lambda(m)(R_\gamma(m)-B_\gamma(m))\) and \(\beta_n=b_n\lambda(n)n^{iv}\). The proof of Proposition 5.1 of Lesson 17 shows that \(\mathbf 1_I(m)m^{iv}(R_\gamma(m)-B_\gamma(m))\) satisfies (2.2) of Lesson 17; multiplying by \(\lambda(m)\) keeps this property, because \(\lambda\chi\) is a character of modulus at most \(ML^{A_0}\le L^{A_0+1}\). Both sequences are supported on \(W\)-rough integers and bounded by \(L^2\), since \(x^\gamma>W\), \(D_\gamma\) is bounded on \([s_-,s_+]\) and \(LV(W)\asymp L^{0.76}\). For \(|v|>L\), \(\int_{|v|>L}|\widehat\psi|\le C_AL^{-A}\), while the bilinear sum is \(O(x)\) by (1.3). Integrating over \(v\) and summing the \(\varphi(M)\) characters gives (2.2). \(\square\)

3. The harmonic mass

For group integers \(r\) put \[ \begin{gathered} J_0=\sum_r\frac{\mathcal W(r)}r,\\ A_r=\sum_{\substack{Q>x^{0.9}\ \text{prime}\\ crQ+1\equiv u\ (M)}}\Psi\Bigl(\frac{crQ+1}x\Bigr),\\ X_0=\sum_r\mathcal W(r)A_r .\quad(3.1) \end{gathered} \]

Lemma 3.1 (harmonic mass [OpenAI-029, Lemma 12.2]). With \(E_i(t)=\prod_{p\in\mathcal P_i}\bigl(1+\frac t{p-1}\bigr)\), \[ \begin{aligned} J_0&=\prod_{i=1}^K\frac{E_i'(1/2)}{V_i}\\ &=\prod_{i=1}^K\Bigl\{\frac{E_i(1/2)}{V_i}\sum_{p\in\mathcal P_i}\frac1{p-\frac12}\Bigr\}.\quad(3.2) \end{aligned} \] Moreover, for every fixed \(\epsilon>0\), \[ \begin{gathered} 1\le J_0\le C_K,\\ \begin{aligned} &\sum_{r>x^\epsilon}\frac{\mathcal W(r)}r\\ &\quad \le C_K\exp\bigl(-\epsilon L^{1-a_K}\bigr), \end{aligned}\quad(3.3) \end{gathered} \] and \(A_r\le Cx/r\); so the part of \(X_0\) with \(r>x^\epsilon\) is \(O_K(x\exp(-\epsilon L^{1-a_K}))\).

Proof. Summing \(p^{-j}\) over \(j\ge1\) gives \(1/(p-1)\), so \(E_i(t)=\sum t^{\omega(r_i)}/r_i\) over the integers \(r_i\) supported on \(\mathcal P_i\), and \(E_i'(t)=\sum\omega(r_i)t^{\omega(r_i)-1}/r_i\). With \(q=\frac12\), \(\mathcal W(r)=\prod_i\omega_i(r)2^{1-\omega_i(r)}/V_i\), and a group integer factors uniquely as \(\prod_ir_i\); this gives the first expression in (3.2), and logarithmic differentiation of \(E_i\) gives the second. The integers with one prime, to the first power, from each group contribute \(\prod_i(V_i^{-1}\sum_{p\in\mathcal P_i}p^{-1})=1\), and all terms are nonnegative, so \(J_0\ge1\). Put \(s=L^{-a_K}\). A group prime has \(p\le\exp(2L^{a_K})\), so \(p^s\le e^2\) and \(\sum_{j\ge1}p^{-j(1-s)}\le2e^2/p\) for large \(x\). Bounding \(\mathcal W\) by \(B_K\), \[ \begin{aligned} \sum_r\frac{\mathcal W(r)}{r^{1-s}}&\le B_K\prod_{p\ \text{group}}\Bigl(1+\frac{2e^2}p\Bigr)\\ &\le B_K\exp\Bigl(2e^2\sum_iV_i\Bigr)\\ &\le C_K , \end{aligned} \] which bounds \(J_0\), and on \(r>x^\epsilon\) the factor \(r^{-s}\le x^{-\epsilon s}=\exp(-\epsilon L^{1-a_K})\) gives the tail. Finally \(A_r\) counts at most \(x/(cr)+1\) values of \(Q\). \(\square\)

4. Distribution of the family

Fix parameters \[ \begin{gathered} 0<\kappa<0.01,\quad 0<\epsilon<\kappa,\\ 0<b<0.01,\quad D=x^{1/2-\kappa/2},\quad(4.1) \end{gathered} \] and put \(N=M/c\). For an odd squarefree \(\ell\), let \(A_r(\ell)\) be \(A_r\) restricted to \(\ell\mid crQ+1\), and \[ \begin{gathered} g_r(\ell)=\frac{\mathbf 1\{(\ell,Mr)=1\}}{\varphi(\ell)},\\ E_r(\ell)=A_r(\ell)-g_r(\ell)A_r ,\quad(4.2) \end{gathered} \] so \(g_r(1)=1\), \(E_r(1)=0\).

Lemma 4.1 (distribution [OpenAI-029, Lemma 12.3]). For every fixed \(A>0\), \[ \begin{aligned} &\sum_{r\le x^\epsilon}\mathcal W(r)\sum_{\substack{\ell\le D\\ \ell\ \text{odd squarefree}}}|E_r(\ell)|\\ &\quad\le C_A\,x\,J_0\,L^{-A}.\quad(4.3) \end{aligned} \] Moreover \[ \begin{gathered} X_0\sim\frac{x\,J_0\,A_\Psi}{c\,\varphi(N)\,L}\\ (x\to\infty),\quad(4.4) \end{gathered} \] and in particular \(X_0\asymp xJ_0/L\), with constants depending only on \(M\) and \(\Psi\).

Proof. Classes. For \(r\le x^\epsilon\), \(d=crQ+1\in(x,2x)\) forces \(Q\ge x^{1-\epsilon}/4>x^{0.9}\). The condition \(d\equiv u\pmod M\) is \(rQ\equiv\frac{u-1}c\pmod N\); since \(r\) is prime to \(M\) and \(\frac{u-1}c\) is prime to \(N\) by (1.1), it is one reduced class of \(Q\) modulo \(N\). If an odd prime of \(\ell\) divides \(r\), it cannot divide \(crQ+1\); if it divides \(M\), it cannot divide \(d\equiv u\), as \((u,M)=1\); so \(A_r(\ell)=g_r(\ell)=0\) when \((\ell,Mr)>1\). Otherwise \(\ell\mid crQ+1\) is the reduced class \(Q\equiv-(cr)^{-1}\pmod\ell\), and the two conditions form one reduced class modulo \(N\ell\).

Bombieri–Vinogradov. With \[ \begin{aligned} &E_\pi(Y,k)=\max_{2\le y\le Y}\ \max_{(v,k)=1}\\ &\qquad\Bigl|\pi(y;k,v)-\frac{\operatorname{li}(y)}{\varphi(k)}\Bigr| , \end{aligned} \] Corollary 4.1 of The Bombieri–Vinogradov theorem, (4.2), gives for fixed \(A',\eta>0\) and large \(Y\) \[ \begin{aligned} &\sum_{k<Y^{1/2-\eta}}E_\pi(Y,k)\\ &\quad \le C\,Y(\log Y)^{-A'} ,\quad(4.5) \end{aligned} \] because \(Y^{1/2-\eta}\le\sqrt Y(\log Y)^{-A'-2}\) for large \(Y\). For \(r\le x^\epsilon\) take \(Y=2x/(cr)\ge x^{1-\epsilon}/2\). The moduli \(N\ell\le ND\) lie below \(Y^{1/2-\eta}\) for large \(x\), with \(\eta=(\kappa-\epsilon)/4\), because \[ \begin{aligned} &(1-\epsilon)\Bigl(\frac12-\eta\Bigr)-\Bigl(\frac12-\frac\kappa2\Bigr)\\ &\quad =\frac{(\kappa-\epsilon)(1+\epsilon)}4>0 . \end{aligned} \] Put \(I_r=\frac x{cr}\int_1^2\frac{\Psi(y)\,dy}{\log((xy-1)/(cr))}\); the substitution \(t=(xy-1)/(cr)\) turns it into \(\int\Psi((crt+1)/x)\,dt/\log t\). Write \(A_r(\ell)\) as a Stieltjes integral of \(\Psi((crt+1)/x)\) against \(\pi(t;N\ell,v)\) and integrate by parts. The main term \(\operatorname{li}(t)/\varphi(N\ell)\) returns \(I_r/\varphi(N\ell)\), and the remainder is at most \(E_\pi(Y,N\ell)\int|\Psi'|\), where \(\int|\Psi'|\) does not depend on \(r\). Distinct \(\ell\) give distinct moduli \(N\ell\), so (4.5) gives \[ \begin{aligned} &\sum_{\substack{\ell\le D\\ \ell\ \text{odd squarefree}}}\Bigl|A_r(\ell)-\frac{g_r(\ell)}{\varphi(N)}I_r\Bigr|\\ &\quad\le C_{A'}\frac xrL^{-A'} ,\quad(4.6) \end{aligned} \] since \(g_r(\ell)/\varphi(N)=1/\varphi(N\ell)\) when \((\ell,Mr)=1\), and both terms vanish otherwise. The term \(\ell=1\) gives \(A_r=I_r/\varphi(N)+O((x/r)L^{-A'})\). Replacing \(I_r/\varphi(N)\) by \(A_r\) in the other terms costs \(\sum_{\ell\le D}g_r(\ell)\le\sum_{\ell\le D}1/\varphi(\ell)\le CL\) times this error (expand \(n/\varphi(n)=\sum_{d\mid n}\mu^2(d)/\varphi(d)\)). With \(A'=A+1\) and the weights \(\mathcal W(r)\), (4.3) follows.

The mass. For \(r\le x^\epsilon\) and \(y\in(1,2)\), \(\log((xy-1)/(cr))\asymp L\), so \(I_r\asymp x/(rL)\), and the \(\ell=1\) term and Lemma 3.1 give \(X_0\asymp xJ_0/L\), the part \(r>x^\epsilon\) being negligible since \(J_0\ge1\). For (4.4), fix \(0<\theta<\epsilon\). For \(r\le x^\theta\), \(\log((xy-1)/(cr))=L+O(\theta L+1)\), so \(I_r=\frac{xA_\Psi}{crL}(1+O(\theta)+O(L^{-1}))\) with constants independent of \(\theta\). Summing the \(\ell=1\) evaluation over \(r\le x^\theta\) with weights, and bounding the parts \(r>x^\theta\) of \(X_0\) and of \(J_0\) by Lemma 3.1, gives \(\frac{c\,\varphi(N)\,L\,X_0}{x\,J_0\,A_\Psi}=1+O(\theta)+o_\theta(1)\). Let \(x\to\infty\), then \(\theta\to0\). \(\square\)

For \((M,c,u)=(8,4,5)\), \(N=2\) and (4.4) reads \(X_0\sim xJ_0A_\Psi/(4L)\).

5. The singular product

Put \[ \begin{gathered} V_M(y)=\prod_{\substack{2<p\le y\\ p\nmid M}}\Bigl(1-\frac1{p-1}\Bigr),\\ \begin{aligned} \mathfrak S_M&=2\prod_{p>2}\Bigl(1-\frac1{(p-1)^2}\Bigr)\\ &\quad\times\prod_{\substack{p\mid M\\ p>2}}\Bigl(1-\frac1{p-1}\Bigr)^{-1}>0 .\quad(5.1) \end{aligned} \end{gathered} \]

Lemma 5.1 ([OpenAI-029, (12.10)–(12.11)]). As \(y\to\infty\), \(V_M(y)\sim\mathfrak S_MV(y)\) and \(V(y)\sim e^{-\gamma_E}/\log y\). For group integers \(r\le x^{0.11}\) and all \(y\), \[ \begin{gathered} \sum_{p\mid r}\frac1p\le0.11\,L^{0.9}e^{-L^{0.1}}=o(1),\\ \begin{aligned} &\prod_{2<p\le y}\bigl(1-g_r(p)\bigr)\\ &\quad =(1+o(1))\,V_M(y), \end{aligned}\quad(5.2) \end{gathered} \] uniformly, with an \(o(1)\) independent of \(K\).

Proof. Compare factors with \(V(y)\): the prime \(2\) contributes \(2\); an odd \(p\nmid M\) contributes \(\frac{1-1/(p-1)}{1-1/p}=1-\frac1{(p-1)^2}\); an odd \(p\mid M\) contributes \((1-1/p)^{-1}=\bigl(1-\frac1{(p-1)^2}\bigr)\bigl(1-\frac1{p-1}\bigr)^{-1}\). The product of \(1-(p-1)^{-2}\) converges, which gives \(V_M\sim\mathfrak S_MV\); Mertens' product theorem (Theorem 4.2 of Counting primes by elementary means) gives the second relation. A group integer \(r\le x^{0.11}\) has at most \(0.11L/L^{0.1}\) prime factors, each above \(e^{L^{0.1}}\); this gives the first part of (5.2), and the product differs from \(V_M(y)\) by \(\prod_{p\mid r,\,p\le y}(1-\frac1{p-1})^{-1}=1+o(1)\). \(\square\)

6. Exercises

6.1. Show that \(j2^{1-j}\le1\) for every integer \(j\ge1\), and compute \(\mathcal W(r)\) for \(r=p_1p_2\cdots p_K\) with one prime \(p_i\) from each group.

6.2. For \((M,c,u)=(8,4,5)\), describe the class of \(Q\) modulo \(N=2\) and check that every odd prime \(Q\) is allowed.

6.3. Verify the identity \((1-\epsilon)(\frac12-\eta)-(\frac12-\frac\kappa2)=\frac{(\kappa-\epsilon)(1+\epsilon)}4\) for \(\eta=\frac{\kappa-\epsilon}4\).

6.4. Show that \(\sum_{\ell\le D}1/\varphi(\ell)\le C\log D\) from \(\frac n{\varphi(n)}=\sum_{d\mid n}\frac{\mu^2(d)}{\varphi(d)}\).

6.5. Compute \(\mathfrak S_8\).

7. Solutions

6.1. \(j2^{1-j}\) is \(1\) for \(j=1,2\) and decreases for \(j\ge2\). For \(r=p_1\cdots p_K\), every \(\omega_i(r)=1\) and \(\mathcal W(r)=\prod_iV_i^{-1}\).

6.2. \(rQ\equiv\frac{5-1}4=1\pmod2\): \(Q\) odd, which every prime \(Q>2\) is.

6.3. The left side is \(\frac\kappa2-\eta-\frac\epsilon2+\epsilon\eta\)\(=\frac{2\kappa-\kappa+\epsilon-2\epsilon}4+\frac{\epsilon(\kappa-\epsilon)}4\)\(=\frac{(\kappa-\epsilon)(1+\epsilon)}4\).

6.4. \(\sum_{\ell\le D}\frac1{\varphi(\ell)}=\sum_{\ell\le D}\frac1\ell\sum_{d\mid\ell}\frac{\mu^2(d)}{\varphi(d)}\)\(\le\sum_d\frac{\mu^2(d)}{d\,\varphi(d)}\sum_{k\le D}\frac1k\), and \(\sum_d\mu^2(d)/(d\varphi(d))\) converges.

6.5. \(M=8\) has no odd prime factor, so \(\mathfrak S_8=2\prod_{p>2}(1-(p-1)^{-2})\approx1.3203\), twice the twin prime constant.

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