This is the seventh thread for the Polymath8b project to obtain new bounds for the quantity

$\displaystyle H_m := \liminf_{n \rightarrow\infty} (p_{n+m} - p_n),$

either for small values of ${m}$ (in particular ${m=1,2}$) or asymptotically as ${m \rightarrow \infty}$. The previous thread may be found here. The currently best known bounds on ${H_m}$ can be found at the wiki page.

The current focus is on improving the upper bound on ${H_1}$ under the assumption of the generalised Elliott-Halberstam conjecture (GEH) from ${H_1 \leq 8}$ to ${H_1 \leq 6}$. Very recently, we have been able to exploit GEH more fully, leading to a promising new expansion of the sieve support region. The problem now reduces to the following:

Problem 1 Does there exist a (not necessarily convex) polytope ${R \subset [0,2]^3}$ with quantities ${0 \leq \varepsilon_1,\varepsilon_2,\varepsilon_3 \leq 1}$, and a non-trivial square-integrable function ${F: {\bf R}^3 \rightarrow {\bf R}}$ supported on ${R}$ such that

• ${R + R \subset \{ (x,y,z) \in [0,4]^3: \min(x+y,y+z,z+x) \leq 2 \},}$
• ${\int_0^\infty F(x,y,z)\ dx = 0}$ when ${y+z \geq 1+\varepsilon_1}$;
• ${\int_0^\infty F(x,y,z)\ dy = 0}$ when ${x+z \geq 1+\varepsilon_2}$;
• ${\int_0^\infty F(x,y,z)\ dz = 0}$ when ${x+y \geq 1+\varepsilon_3}$;

and such that we have the inequality

$\displaystyle \int_{y+z \leq 1-\varepsilon_1} (\int_{\bf R} F(x,y,z)\ dx)^2\ dy dz$

$\displaystyle + \int_{z+x \leq 1-\varepsilon_2} (\int_{\bf R} F(x,y,z)\ dy)^2\ dz dx$

$\displaystyle + \int_{x+y \leq 1-\varepsilon_3} (\int_{\bf R} F(x,y,z)\ dz)^2\ dx dy$

$\displaystyle > 2 \int_R F(x,y,z)^2\ dx dy dz?$

An affirmative answer to this question will imply ${H_1 \leq 6}$ on GEH. We are “within two percent” of this claim; we cannot quite reach ${2}$ yet, but have got as far as ${1.962998}$. However, we have not yet fully optimised ${F}$ in the above problem. In particular, the simplex

$\displaystyle R = \{ (x,y,z) \in [0,2]^3: x+y+z \leq 3/2 \}$

is now available, and should lead to some noticeable improvement in the numerology.

There is also a very slim chance that the twin prime conjecture is now provable on GEH. It would require an affirmative solution to the following problem:

Problem 2 Does there exist a (not necessarily convex) polytope ${R \subset [0,2]^2}$ with quantities ${0 \leq \varepsilon_1,\varepsilon_2 \leq 1}$, and a non-trivial square-integrable function ${F: {\bf R}^2 \rightarrow {\bf R}}$ supported on ${R}$ such that

• ${R + R \subset \{ (x,y) \in [0,4]^2: \min(x,y) \leq 2 \}}$

$\displaystyle = [0,2] \times [0,4] \cup [0,4] \times [0,2],$

• ${\int_0^\infty F(x,y)\ dx = 0}$ when ${y \geq 1+\varepsilon_1}$;
• ${\int_0^\infty F(x,y)\ dy = 0}$ when ${x \geq 1+\varepsilon_2}$;

and such that we have the inequality

$\displaystyle \int_{y \leq 1-\varepsilon_1} (\int_{\bf R} F(x,y)\ dx)^2\ dy$

$\displaystyle + \int_{x \leq 1-\varepsilon_2} (\int_{\bf R} F(x,y)\ dy)^2\ dx$

$\displaystyle > 2 \int_R F(x,y)^2\ dx dy?$

We suspect that the answer to this question is negative, but have not formally ruled it out yet.

For the rest of this post, I will justify why positive answers to these sorts of variational problems are sufficient to get bounds on ${H_1}$ (or more generally ${H_m}$).

— 1. Crude sieve bounds —

Let the notation be as in the Polymath8a paper, thus we have an admissible tuple ${(h_1,\dots,h_k)}$, a residue class ${b\ (W)}$ with ${b+h_i}$ coprime to ${W}$ for all ${i=1,\dots,k}$, and an asymptotic parameter ${x}$ going off to infinity. It will be convenient to use the notation

$\displaystyle \log_x y := \frac{\log y}{\log x}.$

We let ${I}$ be the interval

$\displaystyle I := \{ n \in [x,2x]: n = b\ (W) \}$

and for each fixed smooth compactly supported function ${F: [0,+\infty) \rightarrow {\bf R}}$, we let ${\alpha_F: {\bf Z} \rightarrow {\bf R}}$ denote the divisor sum

$\displaystyle \alpha_F(n) := \sum_{d|n} \mu(d) F(\log_x d).$

We wish to understand the correlation of various products of divisor sums on ${I}$. For instance, in this previous blog post, the asymptotic

$\displaystyle \sum_{n \in I} \prod_{i=1}^k \alpha_{F_i}(n+h_i) \alpha_{G_i}(n+h_i) = \delta^k |I| ( \prod_{i=1}^k \int F'_i(t) G'_i(t)\ dt + o(1) ) \ \ \ \ \ (1)$

was established whenever one has the support condition

$\displaystyle \sum_{i=1}^k S(F_i) + S(G_i) < 1 \ \ \ \ \ (2)$

where ${S(F) := \sup \{ x: F(x) \neq 0 \}}$ is the outer edge of the support of ${F}$, and

$\displaystyle \delta := \frac{W}{\phi(W) \log x}.$

We are now interested in understanding the asymptotics when (2) fails. We have a crude pointwise upper bound:

Lemma 3 Let ${F: [0,+\infty) \rightarrow {\bf R}}$ be a fixed smooth compactly supported function. Then for any natural number ${n}$,

$\displaystyle |\alpha_F(n)| \ll \int_0^\infty [ \prod_{p|n} O( \min( t \log_x p, 1 ) ) ] \frac{dt}{1+|t|^A}$

for any fixed ${A>0}$. More generally, for any fixed number ${F_1,\ldots,F_j: [0,+\infty) \rightarrow {\bf R}}$ of fixed smooth compactly supported functions, one has

$\displaystyle |\alpha_{F_1}(n)| \dots |\alpha_{F_j}(n)| \ll \int_0^\infty [ \prod_{p|n} O( \min( t \log_x p, 1 ) ) ] \frac{dt}{1+t^A} \ \ \ \ \ (3)$

Proof: We extend ${F}$ smoothly to all of ${{\bf R}}$ as a compactly supported function, and write the Fourier expansion

$\displaystyle F(x) = \int_{\bf R} \hat F(t) e^{-ixt}\ dt$

for some rapidly decreasing function ${\hat F(t)}$. Then

$\displaystyle \alpha_F(n) = \int_{\bf R} \hat F(t) \sum_{d|n} \mu(d) \exp( - i t \log_x p )\ dt$

$\displaystyle = \int_{\bf R} \hat F(t) \prod_{p|n} (1 - \exp( - i t \log_x p )\ dt.$

Taking absolute values, we conclude that

$\displaystyle |\alpha_F(n)| \leq \int_{\bf R} |\hat F(t)| \prod_{p|n} |1 - \exp( - i t \log_x p|\ dt.$

Since ${|1 - \exp( - i t \log_x p| \ll \min( |t| \log_x p, 1 )}$, the first claim now follows from the rapid decrease of ${\hat F}$. To prove the second claim, we use the first claim to bound the left-hand side of (3) by

$\displaystyle \int_0^\infty \dots \int_0^\infty [\prod_{p|n} O( \prod_{i=1}^j \min( t_i \log_x p, 1 ) )] \frac{dt_1 \dots dt_j}{\prod_{i=1}^j (1+t_i^A)}.$

Bounding

$\displaystyle \prod_{i=1}^j \min( t_i \log_x p, 1 ) \ll \min( (t_1+\dots+t_j) \log_x p, 1)$

and

$\displaystyle \prod_{i=1}^j (1+t_i^A) \gg 1 + (t_1+\dots+t_j)^A$

the claim follows after a change of variables. $\Box$

Lemma 4 For each ${i=1,\dots,k}$, let ${j_i \geq 1}$ and ${m_i \geq 0}$ be fixed, and let ${F_{i,1},\dots,F_{i,j_i}: [0,+\infty) \rightarrow {\bf R}}$ be fixed smooth compactly supported functions. Then

$\displaystyle \sum_{n \in I} \prod_{i=1}^k (\prod_{l=1}^{j_i} |\alpha_{F_l}(n+h_i)| \tau(n+h_i)^{m_i}) \ll \delta^k |I|. \ \ \ \ \ (4)$

We also have the variant

$\displaystyle \sum_{n \in I} \prod_{i=1}^k (\prod_{l=1}^{j_i} |\alpha_{F_l}(n+h_i)| \tau(n+h_i)^{m_i}) 1_{p(n+h_s) \leq x^\epsilon} \ll \epsilon \delta^k |I|. \ \ \ \ \ (5)$

for any ${\epsilon>0}$ and ${s=1,\dots,k}$, where ${p(n)}$ is the least prime factor of ${n}$.

The intuition here is that each of the ${\alpha_{F_l}(n+h_i)}$ is mostly bounded and mostly supported on the ${n}$ for which ${n+h_i}$ is almost prime (so in particular ${\tau(n+h_i)}$ is bounded), which has a density of about ${\delta}$ in ${I}$.

Proof: From (3) (and bounding ${\tau(n)^{m_i} \leq \prod_{p|n} O(1)}$), we can bound the left-hand side of (4) by

$\displaystyle \int_0^\infty \dots \int_0^\infty \sum_{n \in I} \prod_{i=1}^k \prod_{p_i|n+h_i} O( \min(t_i \log_x p_i, 1) ) \frac{dt_1 \dots dt_k}{\prod_{i=1}^k (1+t_i^A)}.$

Let ${c>0}$ be a small fixed number (${c=1/10k}$ will do). For each ${n+h_i}$, we let ${p_{i,1} \leq \ldots \leq p_{i,\Omega(n+h_i)}}$ be the prime factors of ${n+h_i}$ in increasing order (counting multiplicity), and let ${p_{i,1} \dots p_{i,r_i}}$ be the largest product of consecutive primes factors that is bounded by ${x^c}$. In particular, we see that

$\displaystyle p_{i,1} \dots p_{i,r_i+1} \geq x^c$

and hence

$\displaystyle p_{i,r_i+1} \geq x^{c/(r_i+1)}$

which in particular implies that ${\Omega(n+h_i) = O( r_i)}$. This implies that

$\displaystyle \prod_{p_i|n+h_i} O( \min(t_i \log_x p_i, 1) ) \ll \prod_{j=1}^{r_i} O( \min( t_i \log_x p_{i,j}, 1) ).$

Now observe that ${n+h_i = p_{i,1} \dots p_{i,r_i} m}$, where ${m}$ is ${p_{i,r_i+1}}$-rough (i.e. no prime factors less than ${p_{i,r_i+1}}$). In particular, it is ${x^{c/(r_i+1)}}$-rough. Thus we can bound the left-hand side of (4) by

$\displaystyle \sum_{r_1,\dots,r_k \geq 0} \sum_{p_{i,1} \leq \dots \leq p_{i,r_i} \forall i=1,\dots,k} \int_0^\infty \dots \int_0^\infty$

$\displaystyle (\prod_{i=1}^k \prod_{j=1}^{r_i} O(\min( t_i \log_x p_{i,j}, 1)))$

$\displaystyle \sum_{n \in I} \prod_{i=1}^k 1_{n+h_i = p_{i,1} \dots p_{i,r_i} m; m \hbox{ is } x^{c/(r_i+1)}\hbox{-rough}}$

$\displaystyle \frac{dt_1 \dots dt_k}{\prod_{i=1}^k (1+t_i^A)}.$

By using a standard upper bound sieve (and taking ${c}$ small enough), the quantity

$\displaystyle \sum_{n \in I} \prod_{i=1}^k 1_{n+h_i = p_{i,1} \dots p_{i,r_i} m; m \hbox{ is } x^{c/(r_i+1)}\hbox{-rough}}$

may be bounded by

$\displaystyle \delta^k \frac{|I|}{\prod_{i=1}^k \prod_{j=1}^{r_i} p_{i,j}} \prod_{i=1}^k O( r_i + 1 ).$

Since ${O(r_i+1) \leq \prod_{j=1}^{r_i} O(1)}$, we can thus bound the left-hand side of (4) by

$\displaystyle \delta^k |I| \sum_{r_1,\dots,r_k \geq 0} \sum_{p_{i,1} \leq \dots \leq p_{i,r_i} \forall i=1,\dots,k} \int_0^\infty \dots \int_0^\infty$

$\displaystyle (\prod_{i=1}^k \prod_{j=1}^{r_i} O(\frac{\min( t_i \log_x p_{i,j}, 1)}{p_{i,j}}))$

$\displaystyle \frac{dt_1 \dots dt_k}{\prod_{i=1}^k (1+t_i^A)}.$

We can bound this by

$\displaystyle \delta^k |I| \sum_{r_1,\dots,r_k \geq 0} \int_0^\infty \dots \int_0^\infty$

$\displaystyle (\prod_{i=1}^k \frac{1}{r_i!} O(\sum_{p \leq x^c} \frac{\min( t_i \log_x p, 1)}{p})^{r_i})$

$\displaystyle \frac{dt_1 \dots dt_k}{\prod_{i=1}^k (1+t_i^A)}$

(strictly speaking one has some additional contribution coming from repeated primes ${p_{i,j}=p_{i,j+1}}$, but these can be eliminated in a number of ways, e.g. by restricting initially to square-free ${n+h_i}$). By Mertens’ theorem we have

$\displaystyle \sum_{p \leq x^c} \frac{\min( t_i \log_x p, 1)}{p} = O( 1 + \log(1+t_i) ),$

\endand then by summing the series in ${r_i}$, we can bound the left-hand side of (4) by

$\displaystyle \delta^k |I| \int_0^\infty \dots \int_0^\infty$

$\displaystyle \prod_{i=1}^k \exp( O( 1 + \log(1+t_i) ) )$

$\displaystyle \frac{dt_1 \dots dt_k}{\prod_{i=1}^k (1+t_i^A)}.$

which for ${A}$ large enough is ${O(\delta^k |I|)}$ as required. This proves (4).

The proof of (5) is similar, except that (assuming ${\epsilon}$ small, as we may) ${r_s}$ is forced to be at least ${1}$, and ${\log_x p_{s,1}}$ is at most ${\epsilon}$. From this we may effectively extract an additional factor of ${\min( \epsilon t_s, 1 )}$ (times a loss of ${O(1+\log(1+t_s))}$ due to having to reduce ${r_s!}$ to ${(r_s-1)!}$), which gives rise to the additional gain of ${\epsilon}$. $\Box$

— 2. The generalised Elliott-Halberstam conjecture —

We begin by stating the conjecture ${GEH[\theta]}$ more formally, using (a slightly weaker form of) the version from this paper of Bombieri, Friedlander, and Iwaniec. We use the notation from the Polymath8a paper.

Conjecture 5 (GEH) Let ${x^\epsilon \leq N,M \leq x^{1-\epsilon}}$ for some fixed ${\epsilon>0}$, be such that ${NM \sim x}$, and let ${\alpha, \beta}$ be coefficient sequences at scale ${N,M}$. Then

$\displaystyle \sum_{q \lessapprox x^\theta} \sup_{a \in ({\bf Z}/q{\bf Z})^\times} |\Delta(\alpha * \beta; a\ (q))| \ll x \log^{-A} x$

for any fixed ${A>0}$.

We use GEH to refer to the assertion that ${GEH[\theta]}$ holds for all ${0 < \theta < 1}$. As shown by Motohashi, a modification of the proof of the Bombieri-Vinogradov theorem shows that ${GEH[\theta]}$ is true for ${0 < \theta < 1/2}$. (It is possible that some modification of the arguments of Zhang give some weak version of GEH for some ${\theta}$ slightly above ${1/2}$, but we will not focus on that topic here.)

For our purposes, we will need to apply GEH to functions supported on products of ${r}$ primes for a fixed ${r}$ (generalising the von Mangoldt function, which is the focus of the Elliott-Halberstam conjecture EH). More precisely, we have

Proposition 6 Assume ${GEH[\theta]}$ holds. Let ${r \geq 1}$ and ${\epsilon>0}$ be fixed, let ${\Delta_{r,\epsilon} := \{ (t_1,\dots,t_r) \in [\epsilon,1]^r: t_1 \leq \dots \leq t_r; t_1+\dots+t_r=1\}}$, and let ${F: \Delta_{r,\epsilon} \rightarrow {\bf R}}$ be a fixed smooth function. Let ${\tilde F: {\bf N} \rightarrow {\bf R}}$ be the function defined by setting

$\displaystyle \tilde F(n) := F( \log_n p_i, \dots, \log_n p_r)$

whenever ${n=p_1 \dots p_k}$ is the product of ${r}$ distinct primes ${p_1 < \dots < p_r}$ with ${p_1 \geq x^\epsilon}$ for some fixed ${\epsilon>0}$, and ${\tilde F(n)=0}$ otherwise. Then

$\displaystyle \sum_{q \lessapprox x^\theta} \sup_{a \in ({\bf Z}/q{\bf Z})^\times} |\Delta(\tilde F; a\ (q))| \ll x \log^{-A} x$

for any fixed ${A>0}$.

Remark: it may be possible to get some version of this proposition just from EH using Bombieri’s asymptotic sieve.

Proof: (Sketch) This is a standard partitioning argument (not sure where it appears first, though). We choose a fixed ${A'>0}$ that is sufficiently large depending on ${A}$. We can decompose the primes from ${x^\epsilon}$ to ${x}$ into ${O( \log^{A'+1} x)}$ intervals ${[y, (1+\log^{-A} x) y]}$. This splits ${\tilde F}$ into ${O( \log^{rA'+r} x)}$ pieces, depending on which intervals the ${p_i}$ lie in. The contribution when two primes lie in the same interval, or when the products of the specified intervals touches the boundary of ${[x,2x]}$, can be shown to be negligible by crude divisor function estimates if ${A'}$ is large enough (a similar argument appears in the Polymath8a paper), basically because there are only ${O( \log^{(r-1)A'+r-1} x)}$ such terms, and each one contributes ${O( x \log^{-rA'+O(1)} x)}$ to the total. For the remaining pieces, one can approximate ${\tilde F}$ by a constant, up to errors which can also be shown to be negligible by crude estimates for ${A'}$ large enough (each term contributes ${O( x \log^{-(r+1)A'+O(1)} x)}$), and then ${\tilde F}$ can be modeled by a convolution of ${r}$ coefficient sequences at various scales between ${x^\epsilon}$ and ${x}$, at which point one can use GEH to conclude. $\Box$

Corollary 7 Assume ${GEH[\theta]}$ holds for some ${0 < \theta < 1}$. Let ${r\geq 1}$ and ${\epsilon>0}$ be fixed, let ${F: \Delta_{r,\epsilon} \rightarrow {\bf R}}$ be fixed and smooth, and let ${\tilde F}$ be as in the previous proposition. Let ${\nu_1: {\bf N} \rightarrow {\bf R}}$ be a divisor sum of the form

$\displaystyle \nu_1(n) := \sum_{d_2,\ldots,d_k: d_i|n+h_i \hbox{ for } i=2,\dots,k} \lambda_{d_2,\dots,d_k}$

where ${\lambda_{d_2,\dots,d_k} = O( \tau(d_2\dots d_k)^{O(1)} )}$ are coefficients supported on the range ${d_2 \dots d_k \leq x^\theta}$. Then

$\displaystyle \sum_{n \in I} \tilde F(n+h_1) \nu_1(n) = (\frac{1}{|I|} \sum_{n \in I} \tilde F(n+h_1)) (\sum_{n \in I} \nu_1(n))$

$\displaystyle + O( x \log^{-A} x)$

for any fixed ${A>0}$.

Similarly for permutations of the ${h_1,\dots,h_k}$.

Proof: (Sketch) We can rearrange ${\sum_{n \in I} \tilde F(n+h_1) \nu_1(n)}$ as

$\displaystyle \sum_{d_2,\dots,d_k: d_2 \dots d_k \leq x^\theta} \lambda_{d_2,\dots,d_k} (\sum_{n \in I: n = -h_i\ (d_i) \hbox{ for } i=2,\dots,k} \tilde F(n+h_1) ).$

Using the previous proposition, and the Chinese remainder theorem, we may approximate ${\sum_{n \in I: n = -h_i\ (d_i) \hbox{ for } i=2,\dots,k} \tilde F(n+h_1)}$ by

$\displaystyle \frac{1}{\phi([d_2,\dots,d_k])} \sum_{n \in I} \tilde F(n+h_1)$

plus negligible errors (here we need the crude bounds on ${\lambda_{d_2,\dots,d_k}}$ and some standard bounds on the divisor function), thus

$\displaystyle \sum_{n \in I} \tilde F(n+h_1) \nu_1(n) = (\sum_{n \in I} \tilde F(n+h_1)) (\sum_{d_2,\dots,d_k} \frac{\lambda_{d_2,\dots,d_k}}{\phi([d_2,\dots,d_k])} + O( x \log^{-A} x).$

A similar argument gives

$\displaystyle \sum_{n \in I} \nu_1(n) = |I| (\sum_{d_2,\dots,d_k} \frac{\lambda_{d_2,\dots,d_k}}{\phi([d_2,\dots,d_k])} + O( x \log^{-A} x),$

and the claim follows by combining the two assertions. $\Box$

Next, from Mertens’ theorem one easily verifies that

$\displaystyle \sum_{n \in I} \tilde F(n+h_1) = \delta |I| (\int_{\Delta_{r,\epsilon}} \frac{F(t_1,\dots,t_r)}{t_1 \dots t_r} + o(1))$

where ${\delta := \frac{W}{\phi(W)\log x}}$ is the expected density of primes in ${I}$, and the measure on ${\Delta_{r,\epsilon}}$ is the one induced from Lebesgue measure on the first ${r-1}$ coordinates ${t_1,\dots,t_{r-1}}$. (One could improve the ${o(1)}$ term to ${O(\log^{-A} x)}$ here by using the prime number theorem, but it isn’t necessary for our analysis.)

Applying (1), we thus have

Corollary 8 Assume ${GEH[\theta]}$ holds for some ${0 < \theta < 1}$. Let ${r\geq 1}$ and ${\epsilon>0}$ be fixed, let ${F: \Delta_{r,\epsilon} \rightarrow {\bf R}}$ be fixed and smooth, and let ${\tilde F}$ be as in the previous proposition. For ${i=2,\dots,k}$, let ${F_i,G_i: [0,+\infty) \rightarrow {\bf R}}$ be smooth compactly supported functions with ${\sum_{i=2}^k S(F_i)+S(G_i) < \theta}$. Then

$\displaystyle \sum_{n \in I} \tilde F(n+h_1) \prod_{i=2}^k \alpha_{F_i}(n+h_i) \alpha_{G_i}(n+h_i) = \delta^k |I| (X_1 \dots X_k + o(1))$

where

$\displaystyle X_1 := \int_{\Delta_{r,\epsilon}} \frac{F(t_1,\dots,t_r)}{t_1\dots t_r}$

and

$\displaystyle X_i := \int_0^\infty F'_i(t) G'_i(t)\ dt$

for ${i=2,\dots,k}$.

— 3. Some integration identities —

Lemma 9 Let ${f: [0,a] \rightarrow {\bf R}}$ be a smooth function. Then

$\displaystyle \int_0^a f'(t)^2\ dt = \frac{1}{a} (\partial^{(a)} f(0))^2 + \int_{t+u \leq a; t,u \geq 0}(\partial^{(u)} f'(t))^2 \frac{dt du}{a}$

where ${\partial^{(u)} f(t) := f(t+u) - f(t)}$.

Proof: Making the change of variables ${v := u+t}$, the integral ${\int_{t+u \leq a; t,u \geq 0} (\partial^{(u)} f'(t))^2 \frac{dt du}{a}}$ can be written as

$\displaystyle \int_{0 \leq t \leq v \leq a} (f'(v)-f'(t))^2 \frac{dt dv}{a}$

which by symmetry is equal to

$\displaystyle \frac{1}{2} \int_0^a \int_0^a (f'(v)-f'(t))^2 \frac{dt dv}{a}$

which after expanding out the square and using symmetry is equal to

$\displaystyle \int_0^a f'(t)^2\ dt - \frac{1}{a} (\int_0^a f'(t)\ dt)^2$

and the claim follows from the fundamental theorem of calculus. $\Box$

Iterating this lemma ${k}$ times, we conclude that

$\displaystyle \int_0^a f'(t)^2\ dt = \sum_{i=1}^{k-1} \int_{u_1+\dots+u_i = a; u_1,\dots,u_i \geq 0}$

$\displaystyle \frac{(\partial^{(u_1)} \dots \partial^{(u_i)} f(0))^2}{a(a-u_1) \dots (a-u_1-\dots-u_{i-1})}$

$\displaystyle + \int_{u_1+\dots+u_k+t \leq a; u_1,\dots,u_k,t \geq 0} (\partial^{(u_1)} \dots \partial^{(u_k)} f'(t))^2$

$\displaystyle \frac{dt du_1 \dots du_{k-1}}{a(a-u_1) \dots (a-u_1-\dots-u_{k-1})}$

for any ${k \geq 1}$, where the first integral is integrated using ${du_1 \dots du_{i-1}}$. In particular, discarding the final term (which is non-negative) and then letting ${k \rightarrow \infty}$, we obtain the inequality

$\displaystyle \sum_{i=1}^\infty \int_{u_1+\dots+u_i = a; u_1,\dots,u_i \geq 0} \frac{(\partial^{(u_1)} \dots \partial^{(u_i)} f(0))^2}{a(a-u_1) \dots (a-u_1-\dots-u_{i-1})} \ \ \ \ \ (6)$

$\displaystyle \leq \int_0^a f'(t)^2\ dt.$

In fact we have equality:

Proposition 10 Let ${f: [0,a] \rightarrow {\bf R}}$ be smooth. Then

$\displaystyle \sum_{i=1}^\infty \int_{u_1+\dots+u_i = a; u_1,\dots,u_i \geq 0} \frac{(\partial^{(u_1)} \dots \partial^{(u_i)} f(0))^2}{a(a-u_1) \dots (a-u_1-\dots-u_{i-1})} \ \ \ \ \ (7)$

$\displaystyle = \int_0^a f'(t)^2\ dt.$

In particular, by depolarisation we have

$\displaystyle \sum_{i=1}^\infty \int_{u_1+\dots+u_i = a; u_1,\dots,u_i \geq 0} \frac{\partial^{(u_1)} \dots \partial^{(u_i)} f(0) \partial^{(u_1)} \dots \partial^{(u_i)} g(0)}{a(a-u_1) \dots (a-u_1-\dots-u_{i-1})} \ \ \ \ \ (8)$

$\displaystyle = \int_0^a f'(t)g'(t)\ dt$

for smooth ${f,g: [0,a] \rightarrow {\bf R}}$.

Proof: Let ${\epsilon>0}$ be a small quantity, and write

$\displaystyle X_0 := \int_0^a f'(t)^2\ dt.$

From Lemma 9 we have

$\displaystyle X_0 = Y_1 + Z_1 + X_1$

where

$\displaystyle Y_1 := \frac{1}{a} (\partial^{(a)} f(0))^2$

$\displaystyle Z_1 := \int_{u_1+t \leq a; 0 \leq u_1 \leq \epsilon; t \geq 0} (\partial^{(u_1)} f'(t))^2 \frac{dt du_1}{a}$

$\displaystyle X_1 := \int_{u_1+t \leq a; u_1 \geq \epsilon; t \geq 0} (\partial^{(u_1)} f'(t))^2 \frac{dt du_1}{a}.$

From another application of Lemma 9 we have

$\displaystyle X_1 = Y_2 + Z_2 + X_2$

where

$\displaystyle Y_2 := \int_{u_1+u_2= a; u_1 \geq \epsilon; u_2 \geq 0} (\partial^{(u_1)} \partial^{(u_2)} f(0))^2 \frac{du_1}{a(a-u_1)}$

$\displaystyle Z_2 := \int_{u_1+u_2+t \leq a; u_1 \geq \epsilon; 0 \leq u_2 \leq \epsilon; t \geq 0} (\partial^{(u_1)} \partial^{(u_2)} f'(t))^2 \frac{dt du_1 du_2}{a(a-u_1)}$

$\displaystyle X_2 := \int_{u_1+u_2+t \leq a; u_1,u_2 \geq \epsilon; t \geq 0} (\partial^{(u_1)} \partial^{(u_2)} f'(t))^2 \frac{dt du_1 du_2}{a(a-u_1)}.$

Iterating this, we see that

$\displaystyle X_0 = Y_1+\dots+Y_k + Z_1+\dots+Z_k + X_k$

for any ${k \geq 1}$, where

$\displaystyle Y_i := \int_{u_1+\dots+u_i =a; u_1,\dots,u_{i-1} \geq \epsilon; u_i \geq 0} (\partial^{(u_1)} \dots \partial^{(u_i)} f(0))^2$

$\displaystyle Z_i := \int_{u_1+\dots+u_i+t \leq a; u_1,\dots,u_{i-1} \geq \epsilon; 0 \leq u_i \leq \epsilon; t \geq 0}$

$\displaystyle (\partial^{(u_1)} \dots \partial^{(u_i)} f'(t))^2 \frac{dt du_1 \dots du_i}{a(a-u_1)\dots(a-u_1-\dots-u_{i-1})}$

$\displaystyle X_i := \int_{u_1+\dots+u_i+t \leq a; u_1,\dots,u_i \geq \epsilon; t \geq 0} (\partial^{(u_1)} \dots \partial^{(u_i)} f'(t))^2$

$\displaystyle \frac{dt du_1 \dots du_i}{a(a-u_1)\dots(a-u_1-\dots-u_{i-1})}.$

If ${k > 1/\epsilon}$, then ${X_k}$ vanishes, thus

$\displaystyle X_0 = Y_1+\dots+Y_k + Z_1+\dots+Z_k.$

For ${i \geq 2}$, we may rewrite

$\displaystyle Z_i = \int_{u+t \leq a; 0 \leq u\leq \epsilon; t \geq 0} W_i(t,u)\ dt du \ \ \ \ \ (9)$

where

$\displaystyle W_i(t,u) := \int_{u_1+\dots+u_{i-1} \leq a-t-u; u_1,\dots,u_{i-1} \geq \epsilon}$

$\displaystyle (\partial^{(u_1)} \dots \partial^{(u_{i-1})} f_{t,u}(0))^2\ \frac{du_1 \dots du_{i-1}}{a(a-u_1) \dots (a-u_1-\dots-u_{i-1})}$

and ${f_{t,u}(s) := \partial^{(u)} f(s+t)}$. By Fubini’s theorem, we have

$\displaystyle W_i(t,u) = \int_0^{a-t-u} [\int_{u_1+\dots+u_{i-1} = b; u_1,\dots,u_{i-1} \geq \epsilon}$

$\displaystyle \frac{(\partial^{(u_1)} \dots \partial^{(u_{i-1})} f_{t,u}(0))^2}{a(a-u_1) \dots (a-u_1-\dots-u_{i-2})}] \frac{db}{a-b}.$

Discarding the constraints ${u_1,\dots,u_{i-1} \geq \epsilon}$ and using ${a-b \geq u}$ and ${a \geq b}$, we conclude that

$\displaystyle W_i(t,u) \leq \frac{1}{u} \int_0^{a-t-u} [\int_{u_1+\dots+u_{i-1} = b}$

$\displaystyle \frac{(\partial^{(u_1)} \dots \partial^{(u_{i-1})} f_{t,u}(0))^2}{b(b-u_1) \dots (b-u_1-\dots-u_{i-2})}]\ db.$

Summing over ${2 \leq i \leq k}$ using (6), we see that

$\displaystyle \sum_{i=2}^k W_i(t,u) \leq \frac{1}{u} \int_0^{a-t-u} \int_0^b f_{t,u}'(s)^2\ ds db;$

since ${f_{t,u}' = O_f(1)}$ by the smoothness of ${f}$, we conclude that

$\displaystyle \sum_{i=2}^k W_i(t,u) = O_f(1)$

and thus by (9)

$\displaystyle \sum_{i=2}^k Z_i = O_f(\epsilon).$

Direct computation also shows that ${Z_1 = O_f(\epsilon)}$, hence

$\displaystyle X_0 \leq Y_1+\dots+Y_k+O_f(\epsilon)$

and thus

$\displaystyle X_0 \leq \sum_{i=1}^\infty Y_i + O_f(\epsilon).$

But by the monotone convergence theorem, as ${\epsilon \rightarrow 0}$, ${\sum_{i=1}^\infty Y_i}$ converges to

$\displaystyle \sum_{i=1}^\infty \int_{u_1+\dots+u_i = a; u_1,\dots,u_i \geq 0} \frac{(\partial^{(u_1)} \dots \partial^{(u_i)} f(0))^2}{a(a-u_1) \dots (a-u_1-\dots-u_{i-1})}.$

Thus we can complement (6) with the matching upper bound, giving the claim. $\Box$

We can rewrite the above identity using the following cute identity (which presumably has a name?)

Lemma 11 For any positive reals ${t_1,\dots,t_r}$ with ${r \geq 1}$, one has

$\displaystyle \frac{1}{t_1 \dots t_r} = \sum_{\sigma \in S_r} \frac{1}{\prod_{i=1}^r \sum_{j=i}^r t_{\sigma(j)}}$

where ${\sigma}$ ranges over the permutations of ${\{1,\dots,r\}}$.

Thus for instance

$\displaystyle \frac{1}{t_1} = \frac{1}{t_1}$

$\displaystyle \frac{1}{t_1 t_2} = \frac{1}{(t_1+t_2) t_1} + \frac{1}{(t_1+t_2) t_2}$

$\displaystyle \frac{1}{t_1 t_2 t_3} = \frac{1}{(t_1+t_2+t_3)(t_2+t_3)t_3} + \frac{1}{(t_1+t_2+t_3)(t_2+t_3)t_2}$

$\displaystyle + \frac{1}{(t_1+t_2+t_3)(t_1+t_3)t_3} + \frac{1}{(t_1+t_2+t_3)(t_1+t_3)t_1}$

$\displaystyle + \frac{1}{(t_1+t_2+t_3)(t_1+t_2)t_1} + \frac{1}{(t_1+t_2+t_3)(t_1+t_2)t_2}$

and so forth.

Proof: We induct on ${r}$. The case ${r=1}$ is trivial. If ${r>1}$ and the claim has already been proven for ${r-1}$, then from induction hypothesis one has

$\displaystyle \sum_{\sigma \in S_r: \sigma(1)=l} \frac{1}{\prod_{i=1}^r \sum_{j=i}^r t_{\sigma(j)}}$

$\displaystyle = \frac{1}{(t_1+\dots+t_r) \prod_{1 \leq i \leq r: i \neq l} t_i}$

for each ${l=1,\dots,r}$. Summing over ${l}$, we obtain the claim. $\Box$

Proposition 12 Let ${f,g: [0,a] \rightarrow {\bf R}}$ be smooth. Then

$\displaystyle \sum_{i=1}^\infty \int_{u_1+\dots+u_i = a; 0 \leq u_1 \leq \dots \leq u_i \geq 0} \frac{\partial^{(u_1)} \dots \partial^{(u_i)} f(0) \partial^{(u_1)} \dots \partial^{(u_i)} g(0)}{u_1 \dots u_i}$

$\displaystyle = \int_0^a f'(t) g'(t)\ dt.$

Proof: Average (7) over permutations of the ${u_1,\dots,u_i}$ and use Lemma 11. $\Box$

This gives us a variant of

Corollary 13 Assume ${GEH[\theta]}$ holds for some ${0 < \theta < 1}$. For ${i=1,\dots,k}$, let ${F_i,G_i: [0,+\infty) \rightarrow {\bf R}}$ be smooth compactly supported functions with ${\sum_{i=2}^k S(F_i)+S(G_i) < \theta}$. Then

$\displaystyle \sum_{n \in I} \prod_{i=1}^k \alpha_{F_i}(n+h_i) \alpha_{G_i}(n+h_i) = \delta^k |I| (X_1 \dots X_k + o(1)) \ \ \ \ \ (10)$

where

$\displaystyle X_i := \int_0^1 F'_i(t) G'_i(t)\ dt$

for ${i=1,\dots,k}$.

Proof: Let ${\epsilon>0}$. By (5), we have

$\displaystyle \sum_{n \in I} \prod_{i=1}^k \alpha_{F_i}(n+h_i) \alpha_{G_i}(n+h_i) 1_{p(n+h_1) \leq x^\epsilon} = O(\epsilon \delta^k |I| )$

so by paying a cost of ${O(\epsilon \delta^k |I|)}$, we may restrict to ${n+h_1}$ which are ${x^\epsilon}$-rough, and are thus of the form ${p_1 \dots p_r}$ for some ${r \leq 1/\epsilon}$ and ${x^\epsilon < p_1 \leq \dots \leq p_r}$. For ${n+h_1 = p_1 \dots p_r}$ (restricting to squarefree integers ${n+h_1}$ to avoid technicalities), we have

$\displaystyle \alpha_{F_1}(n+h_1) = \partial^{(\log_{n+h_1} p_1)} \dots \partial^{(\log_{n+h_1}p_r} F_1(0)$

and similarly for ${\alpha_{G_1}(n+h_1)}$. Using this and Corollary 13, we may write the left-hand side of (10) as

$\displaystyle \delta^k |I| ( X_2 \dots X_k \sum_{1 \leq r \leq 1/\epsilon} \int_{\Delta_{r,\epsilon}} \partial^{(t_1)} \dots \partial^{(t_r)} F_1(0) \partial^{(t_1)} \dots \partial^{(t_r)} G_1(0) \frac{dt_1 \dots dt_r}{t_1 \dots t_r}$

$\displaystyle + O(\epsilon) + o(1) ).$

Sending ${\epsilon \rightarrow 0}$ and using dominated convergence and Proposition 11, we obtain the claim. $\Box$

Taking linear combinations, we conclude the usual “denominator” asymptotic

$\displaystyle \sum_{n \in I} (\sum_{d_i|n+h_i \forall i=1,\dots,k} \mu(d_1) \dots \mu(d_k) F(\log_x d_1,\dots,\log_x d_k))^2$

$\displaystyle = \delta^k |I| ( X + o(1) )$

with

$\displaystyle X = \int_{[0,1]^k} F_{1,\dots,k}(t_1,\dots,t_k)^2\ dt_1\dots dt_k$

whenever ${F: [0,+\infty)^k \rightarrow {\bf R}}$ is supported on a polytope ${R}$ (not necessarily convex) with

$\displaystyle R+R \subset \bigcup_{i=1}^k \{ (t_1,\dots,t_k): \sum_{j \neq i} t_j < \theta \},$

and is a finite linear combination of tensor products ${F_1(t_1) \dots F_k(t_k)}$ of smooth compactly supported functions. We use this as a replacement for the denominator estimate in this previous blog post, we obtain the criteria described above.