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Long prime polynomials

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

This lesson proves cancellation in \(\sum_{p\in I}\chi(p)p^{-1+it}\) for primes in an interval of length comparable to a fixed power of \(x\), at every height \(L^{B_0}\le|t|\le x^2\), and for characters of modulus up to a power of \(L\). The proof smooths the interval, writes the smoothed sum as a Mellin integral of \(-D'/D\), and moves the contour into the zero-free strip of A zero-free strip at large height. The second part of the lesson combines this estimate with the results of Lessons 2 and 3 to show that the prime-minus-rough coefficient of The marked Type II estimate has twisted cancellation. This reduces Theorem 2.3 of that lesson to Theorem 2.2.

Prerequisites. A zero-free strip at large height, Theorem 4.1 and (4.1); Primes, divisors, mean squares and rough numbers, Lemmas 1.1, 1.2, 4.1 and 4.2; Logarithmic phases at polynomial scales, Corollary 4.1 and Lemma 4.2; Schwartz functions and Fourier inversion, Section F4; Dirichlet series and Euler products, Lemma A.2.

We keep \(L=\log x\), \(T=\log L\), and write \(D(s,\chi)\) for Dirichlet \(L\)-functions.

1. Prime polynomials

Theorem 1.1 (long prime polynomials). Fix \(0<\tau\le1\), \(C_0>0\) and \(A>0\), and put \(B_0=A+8\). For large \(x\), uniformly for

\[ \begin{gathered} \frac{x^\tau}2\le N\le x^{1/\tau},\qquad q\le L^{C_0},\\ L^{B_0}\le|t|\le x^2, \end{gathered} \]

every Dirichlet character \(\chi\) modulo \(q\) and every interval \(I\subseteq[N,2N]\),

\[ \begin{gathered} \Bigl|\sum_{p\in I}\chi(p)\,p^{-1+it}\Bigr|\le C\,L^{-A},\quad(1.1) \end{gathered} \]

with \(C\) depending on \(\tau,C_0,A\).

Proof. Smoothing. Let \(I=[N\xi_1,N\xi_2]\) with \(1\le\xi_1\le\xi_2\le2\); open or half-open intervals differ by at most two primes, each contributing \(O(1/N)\). Put \(\delta=L^{-A-4}\), fix a smooth nonnegative function \(\phi\) supported in \([-1,1]\) with integral \(1\), and let \(g=\mathbf 1_{[\xi_1,\xi_2]}*\phi_\delta\) with \(\phi_\delta(y)=\delta^{-1}\phi(y/\delta)\). Then \(0\le g\le1\), \(g\) is supported in \([\frac12,3]\), \(g(y)=\mathbf 1_{[\xi_1,\xi_2]}(y)\) unless \(y\) is within \(\delta\) of \(\xi_1\) or \(\xi_2\), and \(|g^{(j)}|\le\int|\phi_\delta^{(j)}|\ll_j\delta^{-j}\). Since \(\Lambda(n)\le\log(3N)\) and \(n\ge N/2\) on the support,

\[ \begin{aligned} &\sum_n\frac{\Lambda(n)}n\bigl|\mathbf 1_I(n)-g(n/N)\bigr|\\ &\quad\le2(2\delta N+1)\frac{2\log(3N)}N\\ &\quad\ll L^{-A-2}.\quad(1.2) \end{aligned} \]

Mellin transform. Let \(\widehat g(z)=\int_0^\infty g(w)w^{z-1}dw\), an entire function. For \(|\operatorname{Re}z|\le1\), \(|\widehat g(z)|\le\int_{1/2}^3\max(w^{-2},1)\,dw\le3\), and \(j\) integrations by parts give

\[ \begin{gathered} \widehat g(z)=\frac{(-1)^j}{z(z+1)\cdots(z+j-1)}\\ \times\int_0^\infty g^{(j)}(w)w^{z+j-1}dw,\\ |\widehat g(u+iv)|\ll_j\delta^{-j}(1+|v|)^{-j}\\ (|u|\le1).\quad(1.3) \end{gathered} \]

Fourier inversion in the variable \(\log w\) (Section F4 of Schwartz functions and Fourier inversion, for the smooth compactly supported function of Exercise 6.2) gives \(g(w)=\frac1{2\pi i}\int_{(c)}\widehat g(z)w^{-z}dz\) for every real \(c\), the integral converging absolutely by (1.3). With \(c=1/L\) and absolute convergence of the Dirichlet series,

\[ \begin{aligned} &\sum_n\Lambda(n)\chi(n)n^{-1+it}g(n/N)\\ &\quad=\frac1{2\pi i}\int_{(1/L)}\widehat g(z)N^z\\ &\qquad\times\Bigl(-\frac{D'}D(1-it+z,\chi)\Bigr)dz.\quad(1.4) \end{aligned} \]

On the line \(\operatorname{Re}z=1/L\), \(|D'/D(1+1/L+iu,\chi)|\)\(\le-\zeta'/\zeta(1+1/L)\le L+C_\zeta\) by (4.1) of the previous lesson, and \(|N^z|\le e^{1/\tau}\).

Truncation. Let \(H_0=L^{A+6}\) and \(j=\lceil A\rceil+5\). By (1.3), the parts of (1.4) with \(|\operatorname{Im}z|>H_0\) are at most a constant times

\[ \begin{gathered} L\,\delta^{-j}\int_{H_0}^\infty v^{-j}\,dv\\ \ll L^{1+j(A+4)-(j-1)(A+6)}\\ =L^{A+7-2j}, \end{gathered} \]

which is \(O(L^{-A-2})\).

Contour shift. Put \(w_0=cT^2/(2L)\), with \(c\) from Theorem 4.1 of the previous lesson for the modulus bound \(L^{C_0}\). For \(z\) in the rectangle \(-w_0\le\operatorname{Re}z\le1/L\), \(|\operatorname{Im}z|\le H_0\), the point \(s=1-it+z\) satisfies

\[ \begin{gathered} 1-w_0\le\operatorname{Re}s\le1+\frac1L,\\ 2\le|t|-H_0\le|\operatorname{Im}s|\le x^2+H_0\le\frac{x^3}2, \end{gathered} \]

because \(|t|\ge L^{A+8}\). By Theorem 4.1 of the previous lesson, \(D(s,\chi)\) has no zero there, \(s\ne1\), and \(|D'/D(s,\chi)|\le CL\). Cauchy's theorem (Lemma A.2 of Dirichlet series and Euler products, for the rectangle, on a neighbourhood of which the integrand is holomorphic) moves the segment \([1/L-iH_0,1/L+iH_0]\) to \([-w_0-iH_0,-w_0+iH_0]\) plus two horizontal edges. The edges have length at most \(2w_0\le1\), and on them \(|\widehat g|\ll_j\delta^{-j}H_0^{-j}\) and \(|N^z|\le e^{1/\tau}\); with the same \(j\) they contribute \(O(L^{-A-2})\). On the new vertical segment,

\[ \begin{gathered} |N^z|=N^{-w_0}\le2\exp\Bigl(-\frac{c\tau T^2}2\Bigr)\\ =2L^{-c\tau T/2}, \end{gathered} \]

because \(\log N\ge\tau L-\log2\); with \(|\widehat g|\le3\), \(|D'/D|\le CL\) and length \(2H_0\), it contributes \(O(L^{A+7-c\tau T/2})\), which is \(O(L^{-A-2})\) for large \(x\).

From \(\Lambda\) to primes. Together with (1.2) we obtain \(|\sum_{n\in I}\Lambda(n)\chi(n)n^{-1+it}|\ll L^{-A-2}\) for every interval \(I\subseteq[N,2N]\). Prime powers \(p^m\in[N,2N]\) with \(m\ge2\) number at most \(\sqrt{2N}\log_2(2N)\), and each contributes at most \(\log(2N)/N\); so they change the sum by \(O(N^{-1/2}L^2)\). Finally, Lemma 4.2 of Logarithmic phases at polynomial scales with \(g(y)=1/\log y\), for which \(\sup+\text{variation}\le2/\log N\le4/(\tau L)\), removes the weight \(\log p\) and proves (1.1). \(\square\)

At heights \(|t|\le L^{B_0}\) no such bound can hold for the principal character: for \(t=0\) the sum over \(N<p\le2N\) is \((\log2+o(1))/\log N\) by Mertens' estimate. In that range the cancellation needed below comes instead from comparing primes with rough integers.

2. The prime coefficient

We return to the setting of The marked Type II estimate: \(\delta>0\) and \(c_*\ge1\) are fixed, \(H_m,H_n\ge x^\delta\) and \(x/c_*\le X=H_mH_n\le c_*x\). Then \(x^\delta\le H_m\le c_*x^{1-\delta}\).

Proposition 2.1. Fix \(C>0\). Let \(I\subseteq[H_m,2H_m]\) be an interval, \(|v|\le L^C\), and let \(\alpha\) be the coefficient (2.5) of Lesson 1. For all fixed \(A_0,B,A>0\), every Dirichlet character \(\chi\) of modulus \(k\le L^{A_0}\) and every real \(t\) with \(|t|\le2XL^B\),

\[ \begin{gathered} \Bigl|\frac1{H_m}\sum_m\alpha_m\chi(m)m^{it}\Bigr|\\ \le C_{\rm tc}L^{-A},\quad(2.1) \end{gathered} \]

where \(C_{\rm tc}\) and the threshold for \(x\) depend only on \(\delta,c_*,C,A_0,B,A\).

Proof. Put \(u=t+v\). By definition,

\[ \begin{gathered} \sum_m\alpha_m\chi(m)m^{it}=\Sigma_1-\Sigma_2,\\ \Sigma_1=\sum_{p\in I}\chi(p)p^{iu},\\ \Sigma_2=\frac1{V(W)}\sum_{\substack{m\in I\\ m\ \text{rough}}}\frac{\chi(m)m^{iu}}{\log m}. \end{gathered} \]

Let \(B_0=A+8\) be the exponent of Theorem 1.1 for the precision \(A\), and put \(B_2=\max(C+2,B_0+1,A)\).

Low frequencies: \(|t|\le L^{B_2}\). Then \(|u|\le2L^{B_2}\). Let \(A'=A+B_2+2\). By (1.2) of Lemma 1.1 of Lesson 2 (with \(\delta_1=\delta/2\)), for every interval \(I'\subseteq I\),

\[ \begin{gathered} \sum_{p\in I'}\chi(p)=\mathbf 1_{\chi\ \text{principal}}\int_{I'}\frac{dy}{\log y}\\ +O(H_mL^{-A'}). \end{gathered} \]

Partial summation against \(y^{iu}\), whose modulus is \(1\) and whose variation on \(I\) is at most \(|u|\), gives

\[ \begin{gathered} \Sigma_1=\mathbf 1_{\chi\ \text{principal}}\int_I\frac{y^{iu}\,dy}{\log y}\\ +O\bigl(H_mL^{-A'}(1+|u|)\bigr). \end{gathered} \]

By Lemma 4.2 of Lesson 2, \(\sum_{m\in I',\,m\ \text{rough}}\chi(m)\)\(=\mathbf 1_{\chi\ \text{principal}}V(W)|I'|\)\(+O(H_mL^{-A'})\) on every \(I'\subseteq I\). Partial summation against \(\phi(y)=y^{iu}/(V(W)\log y)\), with \(\sup|\phi|+V_I(\phi)\ll L^{0.24}(1+|u|)/L\) by Lemma 1.2 of Lesson 2, gives

\[ \begin{gathered} \Sigma_2=\mathbf 1_{\chi\ \text{principal}}\int_I\frac{y^{iu}\,dy}{\log y}\\ +O\bigl(H_mL^{-A'}(1+|u|)\bigr). \end{gathered} \]

The main terms cancel, and \(|\Sigma_1-\Sigma_2|\ll H_mL^{-A'+B_2+1}\le H_mL^{-A}\).

High frequencies: \(L^{B_2}<|t|\le2XL^B\). Then \(|u|\ge|t|-L^C\ge|t|/2\ge L^{B_0}\) and \(|u|\le2|t|\le4c_*xL^B\le x^2\) for large \(x\). Theorem 1.1, with \(\tau=\delta\) and \(N=H_m\), gives \(|\sum_{p\in I'}\chi(p)p^{-1+iu}|\ll L^{-A}\) for every interval \(I'\subseteq I\); partial summation against \(g(y)=y\), with \(\sup+\text{variation}\le3H_m\) on \(I\), gives \(|\Sigma_1|\ll H_mL^{-A}\).

For \(\Sigma_2\), let \(T_r(m)=\sum\mu(d)\) over squarefree \(d\mid m\) composed of primes at most \(W\) with at most \(r=\lceil c_1\log L\rceil\) prime factors. As in the proof of Lemma 4.2 of Lesson 2, for \(c_1\) large in terms of \(A\),

\[ \begin{gathered} \sum_{m\in I}\bigl|\mathbf 1_{m\ \text{rough}}-T_r(m)\bigr|\\ \le2H_m\frac{(\log L)^{r+1}}{(r+1)!}+(W+1)^{r+1}\\ \ll H_mL^{-A-1}. \end{gathered} \]

Since \(1/(V(W)\log m)\le L^{0.24}/(\delta L)\cdot2\), replacing the rough indicator by \(T_r\) changes \(\Sigma_2\) by \(O(H_mL^{-A})\). Expanding \(T_r\) and writing \(m=dz\),

\[ \begin{gathered} \sum_{m\in I}\frac{T_r(m)\chi(m)m^{iu}}{\log m}\\ =\sum_d\mu(d)\chi(d)d^{iu}\sum_{a\bmod k}\chi(a)\\ \times\sum_{\substack{z\in I/d\\ z\equiv a\ (k)}}\frac{z^{iu}}{\log(dz)}. \end{gathered} \]

Each \(d\) satisfies \(d\le W^r=x^{o(1)}\), so \(z\) runs over an interval in \([N',2N']\) with \(N'=H_m/d\in[x^{\delta/2},c_*x]\). Corollary 4.1 and Lemma 4.2 of Logarithmic phases at polynomial scales, with \(\delta_1=\delta/2\), \(\Lambda_1=2\) and the weight \(1/\log(dz)\), bound the inner sum by \(O\bigl((N'/(kL))(1/|u|+x^{-c})\bigr)\), with \(c=c(\delta)>0\). Summing over the \(k\) residues and over \(d\), with \(\sum_dN'\le H_m\prod_{p\le W}(1+1/p)\)\(\ll H_mL^{0.24}\), and multiplying by \(1/V(W)\le2L^{0.24}\),

\[ \begin{gathered} |\Sigma_2|\ll H_mL^{0.48-1}\bigl(L^{-B_2}+x^{-c}\bigr)\\ +H_mL^{-A}\ll H_mL^{-A}. \end{gathered} \]

This proves (2.1) for all \(|t|\le2XL^B\). The constants depend on \(\delta,c_*,C,A_0,A\), and on \(B\) only through the threshold for \(x\). \(\square\)

Corollary 2.2. The coefficient (2.5) of Lesson 1 is supported on rough integers, satisfies \(|\alpha_m|\le2\), and has twisted cancellation in the sense of Definition 2.1 of that lesson, with constants depending only on \(\delta,c_*,C\) and on \(A_0,B,A\). Hence Theorem 2.3 of Lesson 1 follows from Theorem 2.2 there.

Proof. The first two assertions are Exercise 5.5 of Lesson 1, and the third is Proposition 2.1. Since \(|\alpha_m|\le2\le L^C\), all hypotheses of Theorem 2.2 hold. \(\square\)

3. Exercises

6.1. Show that \(g=\mathbf 1_{[\xi_1,\xi_2]}*\phi_\delta\) has the properties used in the proof of Theorem 1.1, including \(|g^{(j)}|\le\delta^{-j}\int|\phi^{(j)}|\).

6.2. Derive the inversion formula \(g(w)=\frac1{2\pi i}\int_{(c)}\widehat g(z)w^{-z}dz\) from Fourier inversion for the smooth compactly supported function \(y\mapsto g(e^y)e^{cy}\).

6.3. Show that \(\sum_{p^m\in[N,2N],\,m\ge2}\Lambda(p^m)p^{-m}\)\(\ll N^{-1/2}\log^2(2N)\).

6.4. Show that \(\sum_{N<p\le2N}1/p=(\log2+o(1))/\log N\), and explain why Theorem 1.1 must exclude bounded \(t\).

6.5. In the low-frequency part of the proof of Proposition 2.1, verify the bound \(\sup|\phi|+V_I(\phi)\ll L^{0.24}(1+|u|)/L\) for \(\phi(y)=y^{iu}/(V(W)\log y)\).

4. Solutions

6.1. Since \(\phi_\delta\ge0\) has integral \(1\) and support \([-\delta,\delta]\), \(g(y)=\int_{\xi_1}^{\xi_2}\phi_\delta(y-w)\,dw\in[0,1]\); it equals \(1\) when \([y-\delta,y+\delta]\subseteq[\xi_1,\xi_2]\) and \(0\) when the two intervals are disjoint. Differentiating under the integral, \(g^{(j)}(y)=\int_{\xi_1}^{\xi_2}\phi_\delta^{(j)}(y-w)\,dw\), whose modulus is at most \(\int|\phi_\delta^{(j)}|=\delta^{-j}\int|\phi^{(j)}|\).

6.2. Put \(h(y)=g(e^y)e^{cy}\), smooth with compact support. Its Fourier transform is \(\int h(y)e^{-i\eta y}dy=\widehat g(c-i\eta)\). Fourier inversion gives \(h(y)=\frac1{2\pi}\int\widehat g(c-i\eta)e^{i\eta y}d\eta\); substitute \(z=c-i\eta\) and \(w=e^y\).

6.3. For each \(m\ge2\) there are at most \((2N)^{1/m}\le\sqrt{2N}\) primes \(p\) with \(p^m\le2N\), and \(m\le\log_2(2N)\). Each term is at most \(\log(2N)/N\).

6.4. Mertens' estimate \(\sum_{p\le y}1/p=\log\log y+M+O(1/\log y)\) gives \(\log\log(2N)-\log\log N\)\(+O(1/\log N)\)\(=\log(1+\log2/\log N)\)\(+O(1/\log N)\). For \(\chi\) principal and \(t=0\) the left side of (1.1) is therefore of exact order \(1/\log N\asymp1/L\), which is not \(O(L^{-A})\) for \(A>1\).

6.5. By Lemma 1.2 of Lesson 2, \(1/V(W)\le2L^{0.24}\), and \(\log y\ge\delta L\) on \(I\). So \(|\phi|\le2L^{0.24}/(\delta L)\), and \(|\phi'(y)|\)\(\le(|u|/y)|\phi(y)|+1/(V(W)y\log^2y)\), whose integral over \(I\subseteq[H_m,2H_m]\) is at most \((|u|+1)\cdot2L^{0.24}/(\delta L)\).

References