Diagonal cubic forms and the large sieve

Victor Y. Wang Fine Hall, 304 Washington Road, Princeton, NJ 08540, USA Courant Institute, 251 Mercer Street, New York, NY 10012, USA IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria vywang@alum.mit.edu

Abstract.

Let $N(X)$ be the number of integral zeros $(x_{1},\dots,x_{6})\in[-X,X]^{6}$ of $\sum_{1\leq i\leq 6}x_{i}^{3}$ . Works of Hooley and Heath-Brown imply $N(X)\ll_{\epsilon}X^{3+\epsilon}$ , if one assumes automorphy and GRH for certain Hasse–Weil $L$ -functions. Assuming instead a natural large sieve inequality, we recover the same bound on $N(X)$ . This is part of a more general statement, for diagonal cubic forms in $\geq 4$ variables, where we allow approximations to Hasse–Weil $L$ -functions.

Key words and phrases:

Cubic form, circle method, rational points, Hasse–Weil

L

-functions, large sieve

1991 Mathematics Subject Classification:

Primary 11D45; Secondary 11D25, 11G40, 11N35, 11P55

1. Introduction

Fix an integer $m\geq 4$ . Fix integers $F_{1},\dots,F_{m}\in\mathbb{Z}\setminus\{0\}$ and let

{\textstyle F(\bm{x})\colonequals\sum_{1\leq i\leq m}F_{i}x_{i}^{3}},

where $\bm{x}=(x_{1},\dots,x_{m})$ . We are interested in the behavior, as $X\to\infty$ , of the point count

N_{F}(X)\colonequals\lvert\{\bm{x}\in\mathbb{Z}^{m}\cap[-X,X]^{m}:F(\bm{x})=0\}\rvert.

Certain varieties, $V_{\bm{c},k}$ , play a key role. For each $\bm{c}=(c_{1},\dots,c_{m})\in\mathbb{Z}^{m}$ and field $k$ , let

{\textstyle V_{\bm{c},k}\colonequals\{(\xi_{1},\dots,\xi_{m})\in\mathbb{P}^{m-1}_{k}:\sum_{1\leq i\leq m}F_{i}\xi_{i}^{3}=\sum_{1\leq i\leq m}c_{i}\xi_{i}=0\}},

where $\mathbb{P}^{m-1}_{k}$ is the projective space with coordinates $\xi_{1},\dots,\xi_{m}$ over $k$ .

In the special case $F=\sum_{1\leq i\leq 6}x_{i}^{3}$ , with $m=6$ , we abbreviate $N_{F}(X)$ to $N(X)$ . In this case, building on [hooley1986HasseWeil], the papers [hooley_greaves_harman_huxley_1997] and [heath1998circle] each proved

(1.1)

N(X)\ll_{\epsilon}X^{3+\epsilon},

assuming Hypothesis HW of [hooley1986HasseWeil]*§6; [heath1998circle]*§4 for the Hasse–Weil $L$ -function of each smooth variety $V_{\bm{c},\mathbb{Q}}$ with $\bm{c}\neq\bm{0}$ . Unconditionally, by [vaughan2020some]*Theorem 1.2,

N(X)\ll_{\epsilon}X^{7/2}/(\log{X})^{5/2-\epsilon}

for $X\geq 2$ , via methods stemming from work such as [vaughan1986waring, hall1988divisors, boklan1993reduction, brudern2010asymptotic].

Hypothesis HW practically amounts to automorphy, plus the Grand Riemann Hypothesis (GRH). Automorphy remains open [wang2022thesis]*Appendix A. Hooley suggests that a zero-density hypothesis would suffice in place of GRH [hooley1986HasseWeil]*p. 51. Following the usual paths laid out in [iwaniec2004analytic]*Theorem 10.4, a general such density hypothesis is provable assuming automorphy, a large sieve inequality, and progress on the Grand Lindelöf Hypothesis (GLH).

In the present paper, we show that a large sieve inequality by itself would imply (1.1). The precise large sieve inequality we need will be stated in §2, as Hypothesis 2.1.

Theorem 1.1.

Suppose $m\in\{5,6\}$ . Assume Hypothesis 2.1. Then

(1.2)

N_{F}(X)\ll_{\epsilon}X^{3(m-2)/4+\epsilon},

for all reals $X\geq 1$ and $\epsilon>0$ .

For $m=6$ , the exponent in (1.2) matches (1.1). In §2, we state a more general result, Theorem 2.7, valid for all $m\geq 4$ . Our methods might also apply elsewhere [wang2022thesis]*§9.1. For instance, [restricted_cubic_moments] explains how one may hope to use the modularity of elliptic curves over $\mathbb{Q}$ to unconditionally produce an absolute constant $\delta>0$ such that

\lvert\{a\in\mathbb{Z}:1\leq a\leq A\}\setminus\{x^{2}+y^{3}+z^{3}:x,y,z\in\mathbb{Z}_{\geq 0}\}\rvert\ll A^{6/7-\delta}.

This would then improve on the existing bound $O_{\epsilon}(A^{6/7+\epsilon})$ due to Brüdern [brudern1991ternary].

Conventions

We let $\mathbb{Z}_{\geq c}\colonequals\{n\in\mathbb{Z}:n\geq c\}$ . We let $\bm{1}_{E}\colonequals 1$ if a statement $E$ holds, and $\bm{1}_{E}\colonequals 0$ otherwise. For integers $n\geq 1$ , we let $\mu(n)$ denote the Möbius function.

We write $f\ll_{S}g$ , or $g\gg_{S}f$ , to mean $\lvert f\rvert\leq Cg$ for some $C=C(S)>0$ . The implied constant $C$ is always allowed to depend on $m$ and $F$ , in addition to $S$ . We let $O_{S}(g)$ denote a quantity that is $\ll_{S}g$ . We write $f\asymp_{S}g$ if $f\ll_{S}g\ll_{S}f$ .

2. Framework and results

Let $\mathfrak{D}\colonequals 3(\prod_{1\leq i\leq m}F_{i})^{2^{m-2}}\in\mathbb{Z}$ . For each $\bm{c}\in\mathbb{Z}^{m}$ , let

(2.1)

\Delta(\bm{c})\colonequals\mathfrak{D}\,\prod_{(\upsilon_{2},\dots,\upsilon_{m})\in\{1,-1\}^{m-1}}\,\biggl{(}(c_{1}^{3}/F_{1})^{1/2}+\sum_{2\leq i\leq m}\upsilon_{i}(c_{i}^{3}/F_{i})^{1/2}\biggr{)}\in\mathbb{Z}.

For each field $k$ in which $\Delta(\bm{c})$ is invertible, the variety $V_{\bm{c},k}$ is a smooth complete intersection, by the Jacobian criterion for smoothness. Let

(2.2)

\mathcal{S}\colonequals\{\bm{c}\in\mathbb{Z}^{m}:\Delta(\bm{c})\neq 0\},\qquad\mathcal{S}(C)\colonequals\mathcal{S}\cap[-C,C]^{m}.

For each $\bm{c}\in\mathcal{S}$ and prime $p$ , we define a local Euler factor $L_{p}(s,\bm{c})$ , following Serre [serre1969facteurs] and Kahn [kahn2020zeta]*§5.6. First, choose a prime $\ell\neq p$ , and let

M(\bm{c},\ell)\colonequals H^{m-3}(V_{\bm{c},\overline{\mathbb{Q}}},\mathbb{Q}_{\ell})/H^{m-3}(\mathbb{P}^{m-1}_{\overline{\mathbb{Q}}},\mathbb{Q}_{\ell}),

where $H^{i}(W,\mathbb{Q}_{\ell})$ denotes the $i$ th $\ell$ -adic cohomology group of $W$ . Let $M(\bm{c},\ell)^{I_{p}}\subseteq M(\bm{c},\ell)$ denote the group of inertia invariants of $M(\bm{c},\ell)$ . Let $\alpha_{\bm{c},j}(p)\in\mathbb{C}$ , for $1\leq j\leq\dim{M(\bm{c},\ell)^{I_{p}}}$ , be the geometric Frobenius eigenvalues on $M(\bm{c},\ell)^{I_{p}}$ . Finally, let

(2.3)

\tilde{\alpha}_{\bm{c},j}(p)\colonequals\frac{\alpha_{\bm{c},j}(p)}{p^{(m-3)/2}},\qquad L_{p}(s,\bm{c})\colonequals\prod_{1\leq j\leq\dim{M(\bm{c},\ell)^{I_{p}}}}(1-\tilde{\alpha}_{\bm{c},j}(p)p^{-s})^{-1}.

On multiplying over $p$ , we obtain for each $\bm{c}\in\mathcal{S}$ a global Hasse–Weil $L$ -function

(2.4)

L(s,\bm{c})\colonequals\prod_{p}L_{p}(s,\bm{c})=\sum_{n\geq 1}\lambda_{\bm{c}}(n)n^{-s},

for some coefficients $\lambda_{\bm{c}}(n)\in\mathbb{C}$ defined by expanding the product over $p$ . We now state Hypothesis 2.1. It asserts a large sieve inequality, (2.5), in a certain range.

Hypothesis 2.1.

For all reals $C,N,\epsilon>0$ with $N\leq C^{3}$ , we have

(2.5)

\sum_{\bm{c}\in\mathcal{S}(C)}\,\Bigl{\lvert}\sum_{n\leq N}v_{n}\,\lambda_{\bm{c}}(n)\Bigr{\rvert}^{2}\ll_{\epsilon}C^{\epsilon}\max(C^{m},N)\,\sum_{n\leq N}\lvert v_{n}\rvert^{2}

for all vectors $(v_{n})_{1\leq n\leq N}\in\mathbb{C}^{\lfloor N\rfloor}$ .

We now make some general comments on $L(s,\bm{c})$ . By [kahn2020zeta]*§5.6.3, §5.6.4 and [laskar2017local]*Corollary 1.2, the factors $L_{p}(s,\bm{c})$ are independent of the choice of $\ell$ , and we have

(2.6)

\lvert\tilde{\alpha}_{\bm{c},j}(p)\rvert\leq 1.

By (2.6), the product and series in (2.4) converge absolutely for $\Re(s)>1$ .

We have $\dim{M(\bm{c},\ell)^{I_{p}}}\leq\dim{M(\bm{c},\ell)}\ll_{m}1$ by [katz2001sums]*Corollary of Theorem 3. Therefore, by (2.6), we have $\lambda_{\bm{c}}(n)\ll_{\epsilon}n^{\epsilon}$ for all $n\geq 1$ . Thus (2.5) is the large sieve inequality that one would naturally expect to hold. In fact, (2.5) could potentially hold in the range $N\leq C^{A}$ for any constant $A>0$ . However, we will only need it in the range $N\leq C^{3}$ .

The coefficients $\lambda_{\bm{c}}(n)$ can be interpreted geometrically, but it would take us too far afield to detail anything but the simplest case. For each $\bm{c}\in\mathbb{Z}^{m}$ and prime $p$ , let

E_{\bm{c}}(p)\colonequals\frac{\lvert\{\bm{x}\in\mathbb{F}_{p}^{m}:F(\bm{x})=\bm{c}\cdot\bm{x}=0\}\rvert-p^{m-2}}{p-1},\qquad E^{\natural}_{\bm{c}}(p)\colonequals\frac{E_{\bm{c}}(p)}{p^{(m-3)/2}},

where $\bm{c}\cdot\bm{x}\colonequals\sum_{1\leq i\leq m}c_{i}x_{i}$ . If $p\nmid\Delta(\bm{c})$ , then $M(\bm{c},\ell)^{I_{p}}=M(\bm{c},\ell)$ and

(2.7)

\lambda_{\bm{c}}(p)=\sum_{1\leq j\leq\dim{M(\bm{c},\ell)}}\tilde{\alpha}_{\bm{c},j}(p)=(-1)^{m-3}E^{\natural}_{\bm{c}}(p),

by (2.3) and the Grothendieck–Lefschetz trace formula.

We emphasize that our $L$ -functions are normalized differently than in [hooley1986HasseWeil, heath1998circle]. If $H(s,\bm{c})$ is the $L$ -function associated to $V_{\bm{c},\mathbb{Q}}$ in [heath1998circle]*§4, then

H(s+\tfrac{m-3}{2},\bm{c})=L(s,\bm{c}).

Proof framework

We will analyze $N_{F}(X)$ using the delta method, due to [duke1993bounds, heath1996new]. This method features some complete exponential sums that we now recall. Let

(2.8)

S_{\bm{c}}(n)\colonequals\sum_{\begin{subarray}{c}1\leq a\leq n:\\ \gcd(a,n)=1\end{subarray}}\,\sum_{1\leq x_{1},\dots,x_{m}\leq n}e^{2\pi i(aF(\bm{x})+\bm{c}\cdot\bm{x})/n},\qquad S^{\natural}_{\bm{c}}(n)\colonequals\frac{S_{\bm{c}}(n)}{n^{(m+1)/2}},

for all $\bm{c}\in\mathbb{Z}^{m}$ and integers $n\geq 1$ . It is known that $S_{\bm{c}}(n)$ is multiplicative in $n$ , meaning that $S_{\bm{c}}(1)=1$ and $S_{\bm{c}}(n_{1}n_{2})=S_{\bm{c}}(n_{1})S_{\bm{c}}(n_{2})$ whenever $\gcd(n_{1},n_{2})=1$ [heath1998circle]*Lemma 4.1. Thus $S^{\natural}_{\bm{c}}(n)$ is also multiplicative in $n$ . For each $\bm{c}\in\mathbb{Z}^{m}$ , let

\Phi(\bm{c},s)\colonequals\sum_{n\geq 1}S^{\natural}_{\bm{c}}(n)n^{-s}=\prod_{p}\Phi_{p}(\bm{c},s),

where $\Phi_{p}(\bm{c},s)\colonequals\sum_{l\geq 0}S^{\natural}_{\bm{c}}(p^{l})p^{-ls}$ . Ultimately, we will see that $S^{\natural}_{\bm{c}}(n)$ is related to $\lambda_{\bm{c}}(n)$ in a way that allows us to apply a large sieve inequality, like (2.5), to the delta method.

Before proceeding, we recall two basic definitions from the theory of Dirichlet series. For $f,g\colon\mathbb{Z}_{\geq 1}\to\mathbb{C}$ , the Dirichlet convolution $f\ast g\colon\mathbb{Z}_{\geq 1}\to\mathbb{C}$ is defined by the formula

(f\ast g)(n)\colonequals\sum_{ab=n}f(a)g(b).

A Dirichlet series $\sum_{n\geq 1}f(n)n^{-s}$ is said to be invertible if $f(1)\neq 0$ , or equivalently, if there exists $g\colon\mathbb{Z}_{\geq 1}\to\mathbb{C}$ with $(f\ast g)(n)=\bm{1}_{n=1}$ .

Our work is based on approximations of Dirichlet series. For each $\bm{c}\in\mathcal{S}$ , let $\Psi(\bm{c},s)$ be an invertible Dirichlet series. The function $\bm{c}\mapsto\Psi(\bm{c},s)$ , from $\mathcal{S}$ to the set of Dirichlet series, will be denoted simply by $\Psi$ . For each $\bm{c}\in\mathcal{S}$ and integer $n\geq 1$ , let

b_{\bm{c}}(n),\,a_{\bm{c}}(n),\,a^{\prime}_{\bm{c}}(n)

be the $n^{-s}$ coefficients of the Dirichlet series

\Psi(\bm{c},s),\,\Psi(\bm{c},s)^{-1},\,\Phi(\bm{c},s)/\Psi(\bm{c},s),

respectively. In terms of Dirichlet convolution, this means that

(2.9)

(a_{\bm{c}}\ast b_{\bm{c}})(n)=\bm{1}_{n=1},\qquad a^{\prime}_{\bm{c}}=S^{\natural}_{\bm{c}}\ast a_{\bm{c}},\qquad S^{\natural}_{\bm{c}}=a^{\prime}_{\bm{c}}\ast b_{\bm{c}}.

For us, the following particular definition of approximation will be convenient.

Definition 2.2.

Call $\Psi$ an approximation of $\Phi$ if the following three conditions hold:

(1)

If $\bm{c}\in\mathcal{S}$ , then $b_{\bm{c}}(n)$ is multiplicative in $n$ .

(2)

For all $\bm{c}\in\mathcal{S}$ , integers $n\geq 1$ , and reals $\epsilon>0$ , we have

\max(\lvert b_{\bm{c}}(n)\rvert,\lvert a^{\prime}_{\bm{c}}(n)\rvert)\ll_{\epsilon}n^{\epsilon}\sum_{d\mid n}\lvert S^{\natural}_{\bm{c}}(d)\rvert.

(3)

For all $\bm{c}\in\mathcal{S}$ and primes $p\nmid\Delta(\bm{c})$ , we have $a^{\prime}_{\bm{c}}(p)\ll p^{-1/2}$ .

Theorem 2.3.

Suppose that for each $\bm{c}\in\mathcal{S}$ , we have

\Psi(\bm{c},s)\in\left\{\Phi(\bm{c},s),\quad\prod_{p\nmid\Delta(\bm{c})}\Phi_{p}(\bm{c},s),\quad\prod_{p\nmid\Delta(\bm{c})}L_{p}(s,\bm{c})^{(-1)^{m-3}},\quad L(s,\bm{c})^{(-1)^{m-3}}\right\}.

Then $\Psi$ is an approximation of $\Phi$ .

Theorem 2.3 provides natural examples of approximations. It will not be used until §8, so we defer the proof to that section. For the rest of §2, fix an approximation $\Psi$ of $\Phi$ .

Hypotheses

Our main general result, Theorem 2.7, will assume that either of two specific hypotheses holds. Our first hypothesis is the following:

Hypothesis 2.4.

For all reals $C,N,\epsilon>0$ with $N\leq C^{3}$ , we have

(2.10)

\sum_{\bm{c}\in\mathcal{S}(C)}\,\Bigl{\lvert}\sum_{n\in I}b_{\bm{c}}(n)\Bigr{\rvert}^{2}\ll_{\epsilon}C^{\epsilon}\max(C^{m},N)\,N

for all real intervals $I\subseteq(0,N]$ .

The following two remarks may help to clarify the nature of this hypothesis.

(1)

If $\Psi=L(s,\bm{c})^{-1}$ , then Hypothesis 2.4 would easily follow from GRH. On the other hand, if $\Psi=L(s,\bm{c})$ , then Hypothesis 2.4 would follow from GLH plus a technical bound on $\lvert\{\bm{c}\in\mathcal{S}(C):L(s,\bm{c})\;\textnormal{has a pole at}\;s=1\}\rvert$ .
(2)

A density bound, namely $\lvert\{\bm{c}\in\mathcal{S}(C):\lvert\sum_{n\in I}b_{\bm{c}}(n)\rvert\geq N^{\sigma}\}\rvert\ll_{\epsilon}C^{m+\epsilon}/N^{2\sigma-1}$ for $N\leq C^{3}$ and $\sigma\geq 1/2$ , would follow from Hypothesis 2.4. But $C^{m+\epsilon}/N^{2\sigma-1}$ could be quite large even if $N=C^{3}$ and $\sigma=1$ . This is unlike in some density applications, e.g. [iwaniec2004analytic]*Theorem 10.5, where further input may be needed near $\sigma=1$ .

If $\Psi=L(s,\bm{c})^{-1}$ , then Hypothesis 2.4 is perhaps unattractive in that $b_{\bm{c}}(n)$ involves the Möbius function $\mu(n)$ . We might thus wish to pass from $b_{\bm{c}}(n)$ to $a_{\bm{c}}(n)$ . This is possible, to some extent, in the situation of the following definition:

Definition 2.5.

Call $\Psi$ standard if for all $\bm{c}\in\mathcal{S}$ , integers $n\geq 1$ , and reals $\epsilon>0$ , we have

\max(\lvert b_{\bm{c}}(n)\rvert,\lvert a_{\bm{c}}(n)\rvert)\ll_{\epsilon}n^{\epsilon}.

Let $\vartheta\in\{0,1\}$ if $\Psi$ is standard, and let $\vartheta\colonequals 0$ if $\Psi$ is non-standard. Let

(2.11)

\gamma_{\bm{c}}(n)\colonequals(1-\vartheta)\cdot b_{\bm{c}}(n)+\vartheta\cdot\mu(n)^{2}a_{\bm{c}}(n).

We now come to our main hypothesis: a large sieve inequality for $\gamma_{\bm{c}}$ , in a certain range.

Hypothesis 2.6.

For all reals $C,N,\epsilon>0$ with $N\leq C^{3}$ , we have

(2.12)

\sum_{\bm{c}\in\mathcal{S}(C)}\,\Bigl{\lvert}\sum_{n\leq N}v_{n}\,\gamma_{\bm{c}}(n)\Bigr{\rvert}^{2}\ll_{\epsilon}C^{\epsilon}\max(C^{m},N)\,\sum_{n\leq N}\lvert v_{n}\rvert^{2}

for all vectors $(v_{n})_{1\leq n\leq N}\in\mathbb{C}^{\lfloor N\rfloor}$ .

Again, some brief remarks may be helpful.

(1)

When $\vartheta=1$ , the factor $\mu(n)^{2}$ in (2.11) simply restricts us to square-free moduli $n$ .
(2)

Hypothesis 2.6 remains open in general [wang2022thesis]*Remark 4.1.10.

Results

Fix a smooth, compactly supported function $w\colon\mathbb{R}^{m}\to\mathbb{R}$ . Assume that

(2.13)

\bm{0}\notin\overline{\{\bm{x}\in\mathbb{R}^{m}:w(\bm{x})\neq 0\}}.

For reals $X\geq 1$ , let

(2.14)

N_{F,w}(X)\colonequals\sum_{\bm{x}\in\mathbb{Z}^{m}}w(\bm{x}/X)\,\bm{1}_{F(\bm{x})=0}.

If $m\geq 5$ , then let $N^{\prime}_{F,w}(X)\colonequals N_{F,w}(X)$ . If $m=4$ , then let $\Upsilon$ denote the set of $2$ -dimensional rational vector spaces $L$ with $F|_{L}=0$ , and let

(2.15)

N^{\prime}_{F,w}(X)\colonequals\sum_{\bm{x}\in\mathbb{Z}^{m}\setminus(\bigcup_{L\in\Upsilon}L)}w(\bm{x}/X)\,\bm{1}_{F(\bm{x})=0}.

Theorem 2.7.

Assume Hypothesis 2.6 or Hypothesis 2.4. Then for some constant $\mathfrak{c}(F,w)\in\mathbb{R}$ , we have

(2.16)

N^{\prime}_{F,w}(X)-\mathfrak{c}(F,w)X^{m-3}\ll_{\epsilon}X^{3(m-2)/4+\epsilon},

for all reals $X\geq 1$ and $\epsilon>0$ .

Note that $m,F,w$ are fixed. In other words, the implied constant in (2.16) is allowed to depend on $m,F,w$ in addition to $\epsilon$ . Also, for numerical reference,

3(m-2)/4=1.5\cdot\bm{1}_{m=4}+2.25\cdot\bm{1}_{m=5}+3\cdot\bm{1}_{m=6}+\cdots.

In particular, if $5\leq m\leq 6$ , then $m-3\leq 3(m-2)/4$ , and (2.16) simply says

N_{F,w}(X)\ll_{\epsilon}X^{3(m-2)/4+\epsilon}.

The rest of the paper is devoted to the proof of Theorems 1.1, 2.3, and 2.7. In §3, we reduce Hypothesis 2.4 to Hypothesis 2.6. In §§4–7, we recall the delta method for $N_{F,w}(X)$ , then analyze parts of it unconditionally and parts of it using Hypothesis 2.4. In §8, we tie together the previous sections to complete the proofs.

3. A conversion between standard coefficients

In this section, we prove a useful consequence of Hypothesis 2.6. First, we record some standard lemmas that will be repeatedly used throughout the paper.

Lemma 3.1.

Let $N,h\in\mathbb{Z}_{\geq 1}$ . Then there are at most $O_{h}(N^{1/h})$ integers $n\in[N,2N)$ such that $v_{p}(n)\geq h$ holds for all primes $p\mid n$ .

Proof.

This is classical; see e.g. [bateman1958theorem]. ∎

To proceed, we need to introduce some notation. We write $u\mid v^{\infty}$ if there exists $k\in\mathbb{Z}_{\geq 1}$ with $u\mid v^{k}$ . For an integer $c\neq 0$ , we let $\operatorname{sq}(c)$ (resp. $\operatorname{cub}(c)$ ) denote the largest square-full (resp. cube-full) positive integer divisor of $c$ . We also let $\operatorname{sq}(0)\colonequals 0$ .

Lemma 3.2.

Let $N,R\in\mathbb{Z}_{\geq 1}$ . Then there are at most $O_{\epsilon}(N^{\epsilon}R^{\epsilon})$ positive integers $n\leq N$ with $n\mid R^{\infty}$ .

Proof.

We have $\sum_{n\mid R^{\infty}}\bm{1}_{n\leq N}\leq\sum_{n\mid R^{\infty}}(N/n)^{\epsilon}=N^{\epsilon}\prod_{p\mid R}(1-p^{-\epsilon})^{-1}\ll_{\epsilon}N^{\epsilon}R^{\epsilon}$ . ∎

Lemma 3.3.

Let $N\in\mathbb{Z}_{\geq 1}$ . Then the following hold:

(1)

We have

$\sum_{n\leq N:\,n=\operatorname{sq}(n)}n^{-1/2}\ll_{\epsilon}N^{\epsilon}.$
(2)

We have

$\sum_{\lvert c\rvert\leq N}\operatorname{sq}(c)^{1/2}\ll_{\epsilon}N^{1+\epsilon}.$
(3)

For any $t\in\mathbb{R}$ , we have

$\sum_{1\leq n\leq N}\operatorname{cub}(n)^{t}\ll_{t,\epsilon}N^{\epsilon}\max(N,N^{1/3+t}).$

Proof.

(1): By the $h=2$ case of Lemma 3.1 in dyadic intervals $n\in[2^{k},2^{k+1})$ , we have

\sum_{n\leq N:\,n=\operatorname{sq}(n)}n^{-1/2}\ll\sum_{0\leq k\leq\log_{2}{N}}(2^{k})^{1/2}(2^{k})^{-1/2}\ll_{\epsilon}N^{\epsilon}.

(2): There are at most $N/d$ positive integers $n\leq N$ with $\operatorname{sq}(n)=d$ . Therefore,

\sum_{\lvert c\rvert\leq N}\operatorname{sq}(c)^{1/2}=2\sum_{1\leq n\leq N}\operatorname{sq}(n)^{1/2}\leq 2\sum_{d\leq N:\,d=\operatorname{sq}(d)}\frac{N}{d}\cdot d^{1/2}\ll_{\epsilon}N^{1+\epsilon},

where the last inequality follows from (1).

(3): There are at most $N/n_{3}$ positive integers $n\leq N$ with $\operatorname{cub}(n)=n_{3}$ . Thus

\begin{split}\sum_{1\leq n\leq N}\operatorname{cub}(n)^{t}&\leq\sum_{n_{3}\leq N:\,n_{3}=\operatorname{cub}(n_{3})}\frac{N}{n_{3}}\cdot n_{3}^{t}\\ &\ll_{t}\sum_{0\leq k\leq\log_{2}{N}}(2^{k})^{1/3}(2^{k})^{t-1}N\ll_{t,\epsilon}N^{\epsilon}\max(N,N^{1/3+t}),\end{split}

by the $h=3$ case of Lemma 3.1 in dyadic intervals $n_{3}\in[2^{k},2^{k+1})$ . ∎

Proposition 3.4.

Fix an approximation $\Psi$ of $\Phi$ . Assume Hypothesis 2.6. Then Hypothesis 2.4 holds.

Proof.

First suppose $\vartheta=0$ . Then $\gamma_{\bm{c}}=b_{\bm{c}}$ by (2.11). For $C,N,I$ as in Hypothesis 2.4, the bound (2.12) with $v_{n}\colonequals\bm{1}_{n\in I}$ thus trivially implies (2.10), as desired.

Now suppose $\vartheta=1$ . Then in particular, $\Psi$ is standard. For the rest of the proof, let $C,d,N$ denote positive variables. For integers $d$ and intervals $I$ , let

A_{\bm{c}}(d,I)\colonequals\sum_{n\in I}\bm{1}_{\gcd(d,n)=1}\,\mu(n)a_{\bm{c}}(n).

We have $\gamma_{\bm{c}}(n)=\mu(n)^{2}a_{\bm{c}}(n)$ by (2.11). Taking $v_{n}\colonequals\bm{1}_{n\in I}\bm{1}_{\gcd(d,n)=1}\,\mu(n)$ in (2.12), and observing that $\mu(n)^{3}=\mu(n)$ , we find that Hypothesis 2.6 implies

(3.1)

\sum_{\bm{c}\in\mathcal{S}(C)}\lvert A_{\bm{c}}(d,I)\rvert^{2}\ll_{\epsilon}C^{\epsilon}\max(C^{m},N)\,N

uniformly over reals $C$ , integers $d$ , reals $N\leq C^{3}$ , and real intervals $I\subseteq(0,N]$ .

To proceed, we rewrite $b_{\bm{c}}(n)$ using multiplicativity. First, by (2.9), for primes $p$ we have

b_{\bm{c}}(p)=-a_{\bm{c}}(p).

Furthermore, an integer $n\geq 1$ can be uniquely expressed in the form $n_{1}d$ , where $d$ is square-full, $n_{1}$ is coprime to $d$ , and $n_{1}$ is square-free. Therefore, for all $n\geq 1$ , we have

(3.2)

b_{\bm{c}}(n)=\sum_{n_{1}d=n}\bm{1}_{\gcd(d,n_{1})=1}\,\mu(n_{1})a_{\bm{c}}(n_{1})\cdot\bm{1}_{d=\operatorname{sq}(d)}\,b_{\bm{c}}(d).

We note here that $\mu(n_{1})$ is supported on square-free integers $n_{1}$ .

Consider a real $C$ , a real $N\leq C^{3}$ , and a real interval $I\subseteq(0,N]$ . Let $B_{\bm{c}}(I)\colonequals\sum_{n\in I}b_{\bm{c}}(n)$ . By (3.2), we have

\begin{split}B_{\bm{c}}(I)&=\sum_{n_{1}d\in I}\bm{1}_{\gcd(d,n_{1})=1}\,\mu(n_{1})a_{\bm{c}}(n_{1})\cdot\bm{1}_{d=\operatorname{sq}(d)}\,b_{\bm{c}}(d)\\ &=\sum_{d\leq N:\,d=\operatorname{sq}(d)}b_{\bm{c}}(d)\cdot A_{\bm{c}}(d,I/d).\end{split}

By the Cauchy–Schwarz inequality over $d$ , it follows that

(3.3)

\begin{split}\sum_{\bm{c}\in\mathcal{S}(C)}\lvert B_{\bm{c}}(I)\rvert^{2}&\leq\sum_{\bm{c}\in\mathcal{S}(C)}\,\biggl{(}\,\sum_{d\leq N:\,d=\operatorname{sq}(d)}\lvert b_{\bm{c}}(d)\rvert^{2}d^{-1/2}\biggr{)}\biggl{(}\,\sum_{d\leq N:\,d=\operatorname{sq}(d)}d^{1/2}\lvert A_{\bm{c}}(d,I/d)\rvert^{2}\biggr{)}\\ &\ll_{\epsilon}N^{\epsilon}\sum_{\bm{c}\in\mathcal{S}(C)}\sum_{d\leq N:\,d=\operatorname{sq}(d)}d^{1/2}\lvert A_{\bm{c}}(d,I/d)\rvert^{2},\end{split}

by Lemma 3.3(1), since $b_{\bm{c}}(d)\ll_{\epsilon}d^{\epsilon}$ by Definition 2.5. Yet for all integers $d$ , we have

\sum_{\bm{c}\in\mathcal{S}(C)}\lvert A_{\bm{c}}(d,I/d)\rvert^{2}\ll_{\epsilon}C^{\epsilon}\max(C^{m},N/d)\,(N/d)

by (3.1), since $N/d\leq N\leq C^{3}$ and $I/d\subseteq(0,N/d]$ . Plugging this into (3.3), we get

\begin{split}\sum_{\bm{c}\in\mathcal{S}(C)}\lvert B_{\bm{c}}(I)\rvert^{2}&\ll_{\epsilon}N^{\epsilon}\sum_{d\leq N:\,d=\operatorname{sq}(d)}d^{1/2}[C^{\epsilon}\max(C^{m},N/d)\,(N/d)]\\ &\ll_{\epsilon}N^{2\epsilon}C^{\epsilon}\max(C^{m},N)\,N,\end{split}

where the second inequality follows from Lemma 3.3(1) and the trivial bound $\max(C^{m},N/d)\leq\max(C^{m},N)$ . Thus (2.10) holds, uniformly over $C,N,I$ . ∎

4. Delta method ingredients

Let $X\geq 1$ . Assume (2.13), i.e. that $w$ is supported away from $\bm{0}\in\mathbb{R}^{m}$ . Such an assumption is implicit in some of the integral estimates in [heath1996new, heath1998circle]. Set

(4.1)

Y\colonequals X^{(\deg F)/2}=X^{3/2}.

Fix $\epsilon_{0}\in(0,10^{-10}]$ and set

(4.2)

Z\colonequals Y/X^{1-\epsilon_{0}}=X^{1/2+\epsilon_{0}}.

Let $\varrho_{0}(x)\colonequals\exp(-(1-x^{2})^{-1})$ for $\lvert x\rvert<1$ , and $\varrho_{0}(x)\colonequals 0$ for $\lvert x\rvert\geq 1$ . Let

\varrho(x)\colonequals\frac{4\varrho_{0}(4x-3)}{\int_{y\in\mathbb{R}}\varrho_{0}(y)\,dy}.

For $x>0$ and $y\in\mathbb{R}$ , let

h(x,y)\colonequals\sum_{j\geq 1}\frac{1}{xj}\left(\varrho(xj)-\varrho{\left(\frac{\lvert y\rvert}{xj}\right)}\right).

This is precisely the function $h(x,y)$ defined in [heath1996new]*§3. For $\bm{c}\in\mathbb{Z}^{m}$ and $n>0$ , let

I_{\bm{c}}(n)\colonequals\int_{\bm{x}\in\mathbb{R}^{m}}w(\bm{x}/X)\,h(n/Y,F(\bm{x})/Y^{2})\,e^{-2\pi i(\bm{c}\cdot\bm{x}/n)}\,d\bm{x}.

Let $\lVert\bm{c}\rVert\colonequals\max_{1\leq i\leq m}(\lvert c_{i}\rvert)$ . We now recall two standard results on the integral $I_{\bm{c}}(n)$ .

Proposition 4.1 ([heath1996new]*par. 1 of §7).

The functions $n\mapsto I_{\bm{c}}(n)$ are supported on a range of the form $n\leq M_{0}(F,w)Y$ , uniformly over $\bm{c}\in\mathbb{Z}^{m}$ , for some constant $M_{0}(F,w)>0$ .

Lemma 4.2 ([heath1998circle]*(3.9)).

If $\lVert\bm{c}\rVert\geq Z$ and $n\geq 1$ , then $I_{\bm{c}}(n)\ll_{\epsilon_{0},A}\lVert\bm{c}\rVert^{-A}$ , for all $A>0$ .

Proposition 4.1 and Lemma 4.2, together with the trivial bound $\lvert S_{\bm{c}}(n)\rvert\leq n^{1+m}$ , imply

(4.3)

Y^{-2}\sum_{n\geq 1}\sum_{\lVert\bm{c}\rVert>Z}n^{-m}\lvert S_{\bm{c}}(n)I_{\bm{c}}(n)\rvert\ll_{\epsilon_{0},A}X^{-A},

for all $A>0$ . Here $S_{\bm{c}}(n)$ is defined as in (2.8). By [heath1996new]*Theorem 2, (1.2), we have

(4.4)

(1+O_{A}(Y^{-A}))\,N_{F,w}(X)=Y^{-2}\sum_{n\geq 1}\sum_{\bm{c}\in\mathbb{Z}^{m}}n^{-m}S_{\bm{c}}(n)I_{\bm{c}}(n).

Equivalently, in terms of $S^{\natural}_{\bm{c}}(n)$ , we have

(4.5)

(1+O_{A}(X^{-A}))\,N_{F,w}(X)=X^{-3}\sum_{n\geq 1}\sum_{\bm{c}\in\mathbb{Z}^{m}}n^{(1-m)/2}S^{\natural}_{\bm{c}}(n)I_{\bm{c}}(n).

In view of (4.3), analyzing $N_{F,w}(X)$ reduces to understanding the quantity

(4.6)

\Sigma_{0}\colonequals X^{-3}\sum_{n\geq 1}\sum_{\bm{c}\in[-Z,Z]^{m}}n^{(1-m)/2}S^{\natural}_{\bm{c}}(n)I_{\bm{c}}(n).

(Here $I_{\bm{c}}(n)=I_{\bm{c}}(n)\,\bm{1}_{n\leq M_{0}(F,w)Y}$ . But it is more convenient to keep the factor $\bm{1}_{n\leq M_{0}(F,w)Y}$ implicit, in order to allow for more flexible technique later on.)

We now recall some standard formulas for $S_{\bm{c}}$ at primes $p$ and prime powers $p^{l}$ .

Proposition 4.3.

Say $p\nmid\bm{c}$ . Then $S^{\natural}_{\bm{c}}(p)=E^{\natural}_{\bm{c}}(p)+O(p^{-1/2})$ .

Proof.

Let

E(p)\colonequals\frac{\lvert\{\bm{x}\in\mathbb{F}_{p}^{m}:F(\bm{x})=0\}\rvert-p^{m-1}}{p-1}.

By [heath1998circle]*p. 680, we have $S_{\bm{c}}(p)=p^{2}E_{\bm{c}}(p)-pE(p)$ and $E(p)\ll p^{(m-2)/2}$ . Thus

S_{\bm{c}}(p)=p^{2}E_{\bm{c}}(p)+O(p^{m/2}).

Now divide by $p^{(m+1)/2}$ . ∎

Proposition 4.4.

Say $p\nmid\Delta(\bm{c})$ . Then $S_{\bm{c}}(p^{l})=0$ for all integers $l\geq 2$ .

Proof.

This follows immediately from [heath1998circle]*Lemma 4.4. ∎

Fix an approximation $\Psi$ of $\Phi$ . Recall the definition of $\mathcal{S}$ from (2.2). For each $\bm{c}\in\mathcal{S}$ , we have $S^{\natural}_{\bm{c}}=a^{\prime}_{\bm{c}}\ast b_{\bm{c}}$ by (2.9). The following result controls the coefficients $a^{\prime}_{\bm{c}}$ and $b_{\bm{c}}$ .

Proposition 4.5.

Let $\bm{c}\in\mathcal{S}$ . Then $a^{\prime}_{\bm{c}}(n)$ is multiplicative in $n$ . Moreover, for all primes $p$ and integers $k\geq 1$ , we have

\begin{split}a^{\prime}_{\bm{c}}(p)\cdot\bm{1}_{p\nmid\Delta(\bm{c})}&\ll p^{-1/2},\\ \max(\lvert a^{\prime}_{\bm{c}}(p^{k})\rvert,\lvert b_{\bm{c}}(p^{k})\rvert)&\ll_{\epsilon}p^{k\epsilon}+p^{k\epsilon}\sum_{d\mid p^{k}}\lvert S^{\natural}_{\bm{c}}(d)\rvert\cdot\bm{1}_{p\mid\Delta(\bm{c})}.\end{split}

Proof.

By (2.9), we have $(a_{\bm{c}}\ast b_{\bm{c}})(n)=\bm{1}_{n=1}$ and $a^{\prime}_{\bm{c}}=S^{\natural}_{\bm{c}}\ast a_{\bm{c}}$ . Since $b_{\bm{c}},S^{\natural}_{\bm{c}}$ are multiplicative, it follows that $a_{\bm{c}},a^{\prime}_{\bm{c}}$ are too. It remains to bound $a^{\prime}_{\bm{c}}(p^{k}),b_{\bm{c}}(p^{k})$ . When $p\mid\Delta(\bm{c})$ , there is nothing to prove, since condition (2) in Definition 2.2 already gives what we want. Now assume $p\nmid\Delta(\bm{c})$ . Then condition (3) in Definition 2.2 gives $a^{\prime}_{\bm{c}}(p)\ll p^{-1/2}$ . On the other hand, $E^{\natural}_{\bm{c}}(p)\ll 1$ by (2.7) and (2.6). Therefore, condition (2) in Definition 2.2 gives

b_{\bm{c}}(p^{k}),a^{\prime}_{\bm{c}}(p^{k})\ll_{\epsilon}p^{k\epsilon}\sum_{d\mid p^{k}}\lvert S^{\natural}_{\bm{c}}(d)\rvert\ll p^{k\epsilon},

because $S^{\natural}_{\bm{c}}(p)=E^{\natural}_{\bm{c}}(p)+O(p^{-1/2})\ll 1$ by Proposition 4.3 and $S^{\natural}_{\bm{c}}(p^{l})\cdot\bm{1}_{l\geq 2}=0$ by Proposition 4.4. This completes the proof. ∎

Let $\omega(n)$ denote the number of distinct prime factors of $n$ . The following result, which is due to [hooley1986HasseWeil, heath1998circle], gives a general pointwise bound on $S^{\natural}_{\bm{c}}(n)$ .

Proposition 4.6.

For some constant $A_{F}>0$ , we have

n^{-1/2}\lvert S^{\natural}_{\bm{c}}(n)\rvert\leq A_{F}^{\omega(n)}\prod_{1\leq i\leq m}\gcd\bigl{(}\operatorname{cub}(n)^{2},\gcd(\operatorname{cub}(n),\operatorname{sq}(c_{i}))^{3}\bigr{)}^{1/12}

for all $\bm{c}\in\mathbb{Z}^{m}$ and integers $n\geq 1$ .

Proof.

By definition, $S^{\natural}_{\bm{c}}(n)=n^{-(m+1)/2}S_{\bm{c}}(n)$ . Moreover, since $F$ is diagonal, we have

S_{\bm{c}}(p^{l})\ll_{F}p^{l(1+m/2)}\prod_{1\leq i\leq m}\gcd\bigl{(}\operatorname{cub}(p^{l})^{2},\gcd(\operatorname{cub}(p^{l}),\operatorname{sq}(c_{i}))^{3}\bigr{)}^{1/12},

by [heath1998circle]*(5.1) and (5.2) for $l\geq 2$ and [heath1983cubic]*Lemma 11 for $l=1$ . The desired result follows immediately from the multiplicativity of $S_{\bm{c}}$ . ∎

We have stated Proposition 4.6 uniformly over $\bm{c}\in\mathbb{Z}^{m}$ . We proceed to analyze the vectors $\bm{c}$ in sets based on which coordinates $c_{i}$ are nonzero. For the rest of §4, we fix a set

(4.7)

\mathcal{I}\subseteq\{1,2,\dots,m\}.

Let

(4.8)

\mathcal{R}\colonequals\{\bm{c}\in\mathbb{Z}^{m}\cap[-Z,Z]^{m}:\bm{1}_{c_{i}\neq 0}=\bm{1}_{i\in\mathcal{I}}\textnormal{ for all $i\in\{1,2,\dots,m\}$}\}.

By definition, if $\bm{c}\in\mathcal{R}$ , then $c_{i}\neq 0$ if and only if $i\in\mathcal{I}$ .

Proposition 4.6 implies that for all $\bm{c}\in\mathcal{R}$ and integers $n\geq 1$ , we have

(4.9)

n^{-1/2}S^{\natural}_{\bm{c}}(n)\ll_{\epsilon}n^{\epsilon}\operatorname{cub}(n)^{(m-\lvert\mathcal{I}\rvert)/6}\prod_{i\in\mathcal{I}}\gcd(\operatorname{cub}(n),\operatorname{sq}(c_{i}))^{1/4}.

We will repeatedly use (4.9) later in the present paper. We now turn to $I_{\bm{c}}(n)$ .

Lemma 4.7 ([heath1996new, heath1998circle]).

Assume $\lvert\mathcal{I}\rvert\geq 1$ . Then uniformly over $\bm{c}\in\mathcal{R}$ , reals $n\geq 1$ , and integers $k\in\{0,1\}$ , we have

n^{k}(\partial/\partial n)^{k}I_{\bm{c}}(n)\ll_{k,\epsilon}X^{m+\epsilon}\left(\frac{X\lVert\bm{c}\rVert}{n}\right)^{1-(m+\lvert\mathcal{I}\rvert)/4}\prod_{i\in\mathcal{I}}\left(\frac{\lVert\bm{c}\rVert}{\lvert c_{i}\rvert}\right)^{1/2}.

Proof.

By [heath1998circle]*Lemma 3.2, since $F$ is diagonal, we have

(4.10)

\begin{split}n^{k}(\partial/\partial n)^{k}I_{\bm{c}}(n)&\ll_{k,\epsilon}\left(\frac{X\lVert\bm{c}\rVert}{n}\right)X^{m+\epsilon}\prod_{1\leq i\leq m}\min\left[\left(\frac{n}{X\lvert c_{i}\rvert}\right)^{1/2},\left(\frac{n}{X\lVert\bm{c}\rVert}\right)^{1/4}\right]\\ &\leq\left(\frac{X\lVert\bm{c}\rVert}{n}\right)X^{m+\epsilon}\prod_{i\in\mathcal{I}}\left(\frac{n}{X\lvert c_{i}\rvert}\right)^{1/2}\prod_{i\notin\mathcal{I}}\left(\frac{n}{X\lVert\bm{c}\rVert}\right)^{1/4}.\end{split}

After writing $(\frac{n}{X\lvert c_{i}\rvert})^{1/2}=(\frac{n}{X\lVert\bm{c}\rVert})^{1/2}\,(\frac{\lVert\bm{c}\rVert}{\lvert c_{i}\rvert})^{1/2}$ in the final line of (4.10), the desired inequality follows from the fact that $1-\lvert\mathcal{I}\rvert/2-(m-\lvert\mathcal{I}\rvert)/4=1-(m+\lvert\mathcal{I}\rvert)/4$ . ∎

For later convenience, we now make a definition: for $\bm{c}\in\mathbb{Z}^{m}$ and integers $N\geq 1$ , let

(4.11)

\lVert I_{\bm{c}}\rVert_{1,\infty;N}\colonequals\sup_{n\in\mathbb{R}:\,N\leq n\leq 4N}\left(\lvert I_{\bm{c}}(n)\rvert+\lvert n(\partial/\partial n)I_{\bm{c}}(n)\rvert\right).

In the rest of §4, we will concern ourselves only with $\bm{c}\in\mathcal{R}$ such that $\Delta(\bm{c})\neq 0$ . If $\lvert\mathcal{I}\rvert=0$ , then no such $\bm{c}$ exist, because $\mathcal{R}=\{\bm{0}\}$ by (4.8). Therefore, we may and do assume $\lvert\mathcal{I}\rvert\geq 1$ for the rest of §4. To proceed further, we break $\mathcal{R}$ into dyadic pieces. For each $i\in\mathcal{I}$ , let $C_{i}\in\{2^{t}:t\in\mathbb{Z}_{\geq 0}\}$ with $1\leq C_{i}\leq Z$ . Write

(4.12)

\mathcal{C}\colonequals\{\bm{c}\in\mathcal{R}:\lvert c_{i}\rvert\in[C_{i},2C_{i})\textnormal{ for all $i\in\mathcal{I}$}\},\qquad C\colonequals\max_{i\in\mathcal{I}}(C_{i}).

Proposition 4.8.

Suppose $N_{0}\in\mathbb{Z}_{\geq 1}$ and $N_{0}\ll X^{O(1)}$ . Then

\sum_{\bm{c}\in\mathcal{C}:\,\Delta(\bm{c})\neq 0}\biggl{(}\,\sum_{n_{0}\in[N_{0},2N_{0})}\lvert a^{\prime}_{\bm{c}}(n_{0})\rvert\biggr{)}^{\!2}\ll_{\epsilon}X^{\epsilon}N_{0}^{1+(m-\lvert\mathcal{I}\rvert)/3}\prod_{i\in\mathcal{I}}C_{i}.

Proof.

Consider an integer $n_{0}\in[N_{0},2N_{0})$ . If $n_{\bm{c}}\colonequals\prod_{p\mid\Delta(\bm{c})}p^{v_{p}(n_{0})}$ and $n_{2}\colonequals\operatorname{sq}(n_{0}/n_{\bm{c}})$ , then Proposition 4.5 implies

\begin{split}a^{\prime}_{\bm{c}}(n_{0})&=a^{\prime}_{\bm{c}}(\tfrac{n_{0}}{n_{\bm{c}}n_{2}})\cdot a^{\prime}_{\bm{c}}(n_{2})\cdot a^{\prime}_{\bm{c}}(n_{\bm{c}})\\ &\ll_{\epsilon}(\tfrac{n_{0}}{n_{\bm{c}}n_{2}})^{-1/2+\epsilon}\cdot n_{2}^{\epsilon}\cdot\lvert a^{\prime}_{\bm{c}}(n_{\bm{c}})\rvert\\ &\leq n_{0}^{-1/2+\epsilon}(n_{\bm{c}}n_{2})^{1/2}\lvert a^{\prime}_{\bm{c}}(n_{\bm{c}})\rvert.\end{split}

Since $n_{\bm{c}}\mid\Delta(\bm{c})^{\infty}$ and $n_{2}$ is square-full, we find, upon summing over $n_{0}$ , that

\begin{split}\sum_{n_{0}\in[N_{0},2N_{0})}\lvert a^{\prime}_{\bm{c}}(n_{0})\rvert&\ll_{\epsilon}\sum_{\begin{subarray}{c}n_{\bm{c}}n_{2}\leq 2N_{0}:\\ n_{\bm{c}}\mid\Delta(\bm{c})^{\infty},\;n_{2}=\operatorname{sq}(n_{2})\end{subarray}}\frac{N_{0}}{n_{\bm{c}}n_{2}}\cdot N_{0}^{-1/2+\epsilon}(n_{\bm{c}}n_{2})^{1/2}\lvert a^{\prime}_{\bm{c}}(n_{\bm{c}})\rvert\\ &\ll_{\epsilon}N_{0}^{1/2+2\epsilon}\sum_{\begin{subarray}{c}n_{\bm{c}}\leq 2N_{0}:\\ n_{\bm{c}}\mid\Delta(\bm{c})^{\infty}\end{subarray}}n_{\bm{c}}^{-1/2}\lvert a^{\prime}_{\bm{c}}(n_{\bm{c}})\rvert\\ &\ll_{\epsilon}N_{0}^{1/2+2\epsilon}(N_{0}C)^{\epsilon}\max_{\begin{subarray}{c}n_{\bm{c}}\leq 2N_{0}:\\ n_{\bm{c}}\mid\Delta(\bm{c})^{\infty}\end{subarray}}n_{\bm{c}}^{-1/2}\lvert a^{\prime}_{\bm{c}}(n_{\bm{c}})\rvert,\end{split}

where we have used Lemma 3.3(1) to sum over $n_{2}\leq 2N_{0}/n_{\bm{c}}$ , and then used Lemma 3.2 to bound the sum over $n_{\bm{c}}$ by a maximum. Furthermore,

\max_{\begin{subarray}{c}n_{\bm{c}}\leq 2N_{0}:\\ n_{\bm{c}}\mid\Delta(\bm{c})^{\infty}\end{subarray}}n_{\bm{c}}^{-1/2}\lvert a^{\prime}_{\bm{c}}(n_{\bm{c}})\rvert\ll_{\epsilon}N_{0}^{2\epsilon}\max_{\begin{subarray}{c}d\leq 2N_{0}:\\ d\mid\Delta(\bm{c})^{\infty}\end{subarray}}d^{-1/2}\lvert S^{\natural}_{\bm{c}}(d)\rvert,

since $a^{\prime}_{\bm{c}}(n_{\bm{c}})\ll_{\epsilon}n_{\bm{c}}^{\epsilon}\sum_{d\mid n_{\bm{c}}}\lvert S^{\natural}_{\bm{c}}(d)\rvert$ by condition (2) in Definition 2.2. But

\begin{split}\sum_{\bm{c}\in\mathcal{C}:\,\Delta(\bm{c})\neq 0}\,\max_{\begin{subarray}{c}d\leq 2N_{0}:\\ d\mid\Delta(\bm{c})^{\infty}\end{subarray}}d^{-1}\lvert S^{\natural}_{\bm{c}}(d)\rvert^{2}&\leq\sum_{\bm{c}\in\mathcal{C}}\max_{d\leq 2N_{0}}d^{-1}\lvert S^{\natural}_{\bm{c}}(d)\rvert^{2}\\ &\ll_{\epsilon}N_{0}^{(m-\lvert\mathcal{I}\rvert)/3+2\epsilon}\sum_{\bm{c}\in\mathcal{C}}\prod_{i\in\mathcal{I}}\operatorname{sq}(c_{i})^{1/2}\end{split}

by (4.9), since $\gcd(\operatorname{cub}(d),\operatorname{sq}(c_{i}))^{1/4}\leq\operatorname{sq}(c_{i})^{1/4}$ . Yet

(4.13)

\sum_{\bm{c}\in\mathcal{C}}\prod_{i\in\mathcal{I}}\operatorname{sq}(c_{i})^{1/2}\ll_{\epsilon}\prod_{i\in\mathcal{I}}C_{i}^{1+\epsilon},

by Lemma 3.3(2). Proposition 4.8 follows upon combining the previous four displays. ∎

We are now prepared to prove a crucial bound for §5.

Lemma 4.9.

Suppose $N_{0},N\in\mathbb{Z}_{\geq 1}$ and $N_{0},N\ll X^{O(1)}$ . Let

Q_{\bm{c}}=\lVert I_{\bm{c}}\rVert_{1,\infty;N}\,\sum_{n_{0}\in[N_{0},2N_{0})}\lvert a^{\prime}_{\bm{c}}(n_{0})\rvert.

Then

\biggl{(}\,\sum_{\bm{c}\in\mathcal{R}:\,\Delta(\bm{c})\neq 0}Q_{\bm{c}}^{2}\biggr{)}^{\!1/2}\ll_{\epsilon}X^{m+\epsilon}N_{0}^{1/2+(m-\lvert\mathcal{I}\rvert)/6}(X/N)^{1-(m+\lvert\mathcal{I}\rvert)/4}\max[Z^{1+(\lvert\mathcal{I}\rvert-m)/4},1].

Proof.

With notation as in Proposition 4.8, consider an element $\bm{c}\in\mathcal{C}$ . Then by (4.12), we have $\lvert c_{i}\rvert\asymp C_{i}$ for all $i\in\mathcal{I}$ , whence $\lVert\bm{c}\rVert\asymp C$ . Now (4.11) and Lemma 4.7 imply

\lVert I_{\bm{c}}\rVert_{1,\infty;N}\ll_{\epsilon}X^{m+\epsilon}(XC/N)^{1-(m+\lvert\mathcal{I}\rvert)/4}\prod_{i\in\mathcal{I}}(C/C_{i})^{1/2},

since $\lvert\mathcal{I}\rvert\geq 1$ . By Proposition 4.8, it follows that

\sum_{\bm{c}\in\mathcal{C}:\,\Delta(\bm{c})\neq 0}Q_{\bm{c}}^{2}\ll_{\epsilon}X^{2m+3\epsilon}N_{0}^{1+(m-\lvert\mathcal{I}\rvert)/3}(XC/N)^{2-(m+\lvert\mathcal{I}\rvert)/2}\prod_{i\in\mathcal{I}}C.

By (4.12) we have $1\leq C\leq Z$ , since $1\leq C_{i}\leq Z$ for all $i$ . The quantity $C^{2-(m+\lvert\mathcal{I}\rvert)/2}\prod_{i\in\mathcal{I}}C=C^{2+(\lvert\mathcal{I}\rvert-m)/2}$ is maximized either at $C=Z$ or $C=1$ , so we conclude that

\sum_{\bm{c}\in\mathcal{C}:\,\Delta(\bm{c})\neq 0}Q_{\bm{c}}^{2}\ll_{\epsilon}X^{2m+3\epsilon}N_{0}^{1+(m-\lvert\mathcal{I}\rvert)/3}(X/N)^{2-(m+\lvert\mathcal{I}\rvert)/2}\max[Z^{2+(\lvert\mathcal{I}\rvert-m)/2},1].

Summing over all possibilities for $\mathcal{C}$ , we get

\sum_{\bm{c}\in\mathcal{R}:\,\Delta(\bm{c})\neq 0}Q_{\bm{c}}^{2}\ll_{\epsilon}X^{2m+4\epsilon}N_{0}^{1+(m-\lvert\mathcal{I}\rvert)/3}(X/N)^{2-(m+\lvert\mathcal{I}\rvert)/2}\max[Z^{2+(\lvert\mathcal{I}\rvert-m)/2},1].

Lemma 4.9 follows upon taking a square root. ∎

Having analyzed $I_{\bm{c}}$ and $a^{\prime}_{\bm{c}}$ above, we now concentrate on $b_{\bm{c}}$ for the rest of §4.

Proposition 4.10.

Let the $C_{i}$ , as well as $\mathcal{C}$ and $C$ , be as specified before Proposition 4.8. Suppose $N_{1}\in\mathbb{Z}_{\geq 1}$ and $N_{1}\ll X^{O(1)}$ . Then

\sum_{\bm{c}\in\mathcal{C}:\,\Delta(\bm{c})\neq 0}\biggl{(}\,\sum_{n_{1}\in[N_{1},2N_{1}]}\lvert b_{\bm{c}}(n_{1})\rvert\biggr{)}^{\!2}\ll_{\epsilon}X^{\epsilon}N_{1}^{\max(2,1+(m-\lvert\mathcal{I}\rvert)/3)}\prod_{i\in\mathcal{I}}C_{i}.

Proof.

We mimic the proof of Proposition 4.8. Consider an integer $n_{1}\in[N_{1},2N_{1}]$ . If $n_{\bm{c}}\colonequals\prod_{p\mid\Delta(\bm{c})}p^{v_{p}(n_{1})}$ , then by Proposition 4.5 and the multiplicativity of $b_{\bm{c}}$ , we have

b_{\bm{c}}(n_{1})=b_{\bm{c}}(n_{1}/n_{\bm{c}})\,b_{\bm{c}}(n_{\bm{c}})\ll_{\epsilon}(n_{1}/n_{\bm{c}})^{\epsilon}\,\lvert b_{\bm{c}}(n_{\bm{c}})\rvert\leq n_{1}^{\epsilon}\,\lvert b_{\bm{c}}(n_{\bm{c}})\rvert.

Upon summing over $n_{1}$ , then,

\begin{split}\sum_{n_{1}\in[N_{1},2N_{1}]}\lvert b_{\bm{c}}(n_{1})\rvert&\ll_{\epsilon}\sum_{n_{\bm{c}}\leq 2N_{1}:\,n_{\bm{c}}\mid\Delta(\bm{c})^{\infty}}\frac{N_{1}}{n_{\bm{c}}}\cdot N_{1}^{\epsilon}\,\lvert b_{\bm{c}}(n_{\bm{c}})\rvert\\ &\ll_{\epsilon}N_{1}^{1+2\epsilon}C^{\epsilon}\max_{n\leq 2N_{1}}n^{-1}\lvert b_{\bm{c}}(n)\rvert\end{split}

by Lemma 3.2. Condition (2) in Definition 2.2 implies

\max_{n\leq 2N_{1}}n^{-1}\lvert b_{\bm{c}}(n)\rvert\ll_{\epsilon}N_{1}^{2\epsilon}\max_{n\leq 2N_{1}}n^{-1}\lvert S^{\natural}_{\bm{c}}(n)\rvert.

But by (4.9), we have

\sum_{\bm{c}\in\mathcal{C}}\max_{n\leq 2N_{1}}n^{-2}\lvert S^{\natural}_{\bm{c}}(n)\rvert^{2}\ll_{\epsilon}N_{1}^{2\epsilon}\max(1,N_{1}^{-1+(m-\lvert\mathcal{I}\rvert)/3})\sum_{\bm{c}\in\mathcal{C}}\prod_{i\in\mathcal{I}}\operatorname{sq}(c_{i})^{1/2}.

The desired result follows upon combining the last three displays with (4.13). ∎

Lemma 4.11.

Suppose $N_{1}\in\mathbb{Z}_{\geq 1}$ and $N_{1}\ll X^{O(1)}$ . Then

\sum_{\bm{c}\in\mathcal{R}:\,\Delta(\bm{c})\neq 0}\biggl{(}\,\sum_{n_{1}\in[N_{1},2N_{1}]}\lvert b_{\bm{c}}(n_{1})\rvert\biggr{)}^{\!2}\ll_{\epsilon}X^{\epsilon}N_{1}^{\max(2,1+(m-\lvert\mathcal{I}\rvert)/3)}Z^{\lvert\mathcal{I}\rvert}.

Proof.

This follows from Proposition 4.10 upon summing over all possibilities for $\mathcal{C}$ . ∎

We need the following lemma in §5. Let

(4.14)

\beta\colonequals 1+10\cdot M_{0}(F,w)\ll 1.

Lemma 4.12.

Assume Hypothesis 2.4. Then

(4.15)

\sum_{\bm{c}\in\mathcal{R}:\,\Delta(\bm{c})\neq 0}\,\Bigl{\lvert}\sum_{n_{1}\in I}b_{\bm{c}}(n_{1})\Bigr{\rvert}^{2}\ll_{\epsilon}\min\left(X^{\epsilon}Z^{m}N_{1},X^{\epsilon}Z^{\lvert\mathcal{I}\rvert}N_{1}^{\max(2,1+(m-\lvert\mathcal{I}\rvert)/3)}\right),

for all positive integers $N_{1}\leq\beta Y$ and real intervals $I\subseteq[N_{1},2N_{1}]$ .

Proof.

The bound $X^{\epsilon}Z^{m}N_{1}$ in (4.15) follows upon applying (2.10) with $C=(2\beta)^{1/3}Z$ and $N=2N_{1}$ . Meanwhile, $X^{\epsilon}Z^{\lvert\mathcal{I}\rvert}N_{1}^{\max(2,1+(m-\lvert\mathcal{I}\rvert)/3)}$ comes from Lemma 4.11. ∎

5. Contribution from smooth hyperplane sections

Recall the key quantity $\Sigma_{0}$ from (4.6), involving a sum over $\bm{c}\in[-Z,Z]^{m}$ . In this section, we concentrate on vectors $\bm{c}\in\mathcal{S}(Z)=\mathcal{S}\cap[-Z,Z]^{m}$ , in the notation of (2.2). Let

\Sigma_{1}\colonequals X^{-3}\sum_{\bm{c}\in\mathcal{S}(Z)}\,\sum_{n\geq 1}n^{(1-m)/2}S^{\natural}_{\bm{c}}(n)I_{\bm{c}}(n).

We will prove the following result:

Theorem 5.1.

Assume Hypothesis 2.4. Then

(5.1)

\Sigma_{1}\ll_{\epsilon_{0}}X^{3(m-2)/4+O(\epsilon_{0})}.

For each $n\geq 1$ , we have $S^{\natural}_{\bm{c}}(n)=\sum_{n_{0}n_{1}=n}a^{\prime}_{\bm{c}}(n_{0})b_{\bm{c}}(n_{1})$ , since $S^{\natural}_{\bm{c}}=a^{\prime}_{\bm{c}}\ast b_{\bm{c}}$ by (2.9). Thus

(5.2)

\Sigma_{1}=X^{-3}\sum_{\bm{c}\in\mathcal{S}(Z)}\,\sum_{n_{0}\geq 1}a^{\prime}_{\bm{c}}(n_{0})\sum_{n_{1}\geq 1}(n_{0}n_{1})^{(1-m)/2}I_{\bm{c}}(n_{0}n_{1})b_{\bm{c}}(n_{1}).

By Proposition 4.1, we have $I_{\bm{c}}(n)=0$ when $n>\beta Y/10$ , where $\beta$ is as in (4.14). Thus

(5.3)

\Sigma_{1}=X^{-3}\sum_{\bm{c}\in\mathcal{S}(Z)}\,\sum_{(N_{0},N_{1})\in\mathcal{A}}\,\Diamond_{\bm{c},N_{0},N_{1}},

where

\begin{split}\mathcal{A}&\colonequals\{(N_{0},N_{1})\in\{2^{t}:t\in\mathbb{Z}_{\geq 0}\}^{2}:N_{0}N_{1}\leq\beta Y/10\},\\ \Diamond_{\bm{c},N_{0},N_{1}}&\colonequals\sum_{n_{0}\in[N_{0},2N_{0})}a^{\prime}_{\bm{c}}(n_{0})\sum_{n_{1}\in[N_{1},2N_{1})}(n_{0}n_{1})^{(1-m)/2}I_{\bm{c}}(n_{0}n_{1})b_{\bm{c}}(n_{1}).\end{split}

For convenience, let $N\colonequals N_{0}N_{1}$ , let $B_{\bm{c}}(J)\colonequals\sum_{n_{1}\in J}b_{\bm{c}}(n_{1})$ for intervals $J$ , and let

\heartsuit_{\bm{c},n_{0},N_{1}}\colonequals\sum_{n_{1}\in[N_{1},2N_{1})}(n_{0}n_{1})^{(1-m)/2}I_{\bm{c}}(n_{0}n_{1})b_{\bm{c}}(n_{1}).

Recall $\lVert I_{\bm{c}}\rVert_{1,\infty;N}$ from (4.11). We now have enough notation to state a key lemma:

Lemma 5.2.

Let $(N_{0},N_{1})\in\mathcal{A}$ . Then there exists a probability measure $\nu=\nu_{N_{0},N_{1}}$ , supported on the real interval $[N_{1},2N_{1}]$ , such that for all $\bm{c}\in\mathcal{S}$ and $n_{0}\in\mathbb{Z}\cap[N_{0},2N_{0})$ , we have

(5.4)

\heartsuit_{\bm{c},n_{0},N_{1}}\ll N^{(1-m)/2}\,\lVert I_{\bm{c}}\rVert_{1,\infty;N}\,\int_{x\in[N_{1},2N_{1}]}\lvert B_{\bm{c}}([N_{1},x))\rvert\,d\nu(x).

Proof.

Let $\bm{c}\in\mathcal{S}$ and $n_{0}\in\mathbb{Z}\cap[N_{0},2N_{0})$ . For brevity, let $I(n)=n^{(1-m)/2}I_{\bm{c}}(n)$ . Then

\heartsuit_{\bm{c},n_{0},N_{1}}=\sum_{n_{1}\in[N_{1},2N_{1})}I(n_{0}n_{1})\cdot b_{\bm{c}}(n_{1}).

By partial summation over $n_{1}$ , it follows that

\begin{split}\lvert\heartsuit_{\bm{c},n_{0},N_{1}}\rvert&\leq\lVert I(r)\rVert_{L^{\infty}([N,4N])}\,\lvert B_{\bm{c}}([N_{1},2N_{1}))\rvert+n_{0}\,\lVert I^{\prime}(r)\rVert_{L^{\infty}([N,4N])}\,\sum_{k\in[N_{1},2N_{1})}\lvert B_{\bm{c}}([N_{1},k))\rvert\\ &\ll\lVert I(r)\rVert_{L^{\infty}([N,4N])}\,\lvert B_{\bm{c}}([N_{1},2N_{1}))\rvert+\frac{N}{N_{1}}\,\lVert I^{\prime}(r)\rVert_{L^{\infty}([N,4N])}\,\sum_{k\in[N_{1},2N_{1})}\lvert B_{\bm{c}}([N_{1},k))\rvert,\end{split}

where $\lVert f(r)\rVert_{L^{\infty}([N,4N])}\colonequals\sup_{r\in[N,4N]}\lvert f(r)\rvert$ for continuous functions $f\colon[N,4N]\to\mathbb{C}$ . Here

\max(\lVert I(r)\rVert_{L^{\infty}([N,4N])},N\,\lVert I^{\prime}(r)\rVert_{L^{\infty}([N,4N])})\ll N^{(1-m)/2}\,\lVert I_{\bm{c}}\rVert_{1,\infty;N}

by (4.11). Finally, let

\nu\colonequals\frac{1}{2}\biggl{(}\delta_{2N_{1}}+\frac{1}{N_{1}}\sum_{k\in[N_{1},2N_{1})}\delta_{k}\biggr{)},

where $\delta_{k}$ is the Dirac measure supported on the singleton set $\{k\}$ . Then $\nu$ is a probability measure supported on $[N_{1},2N_{1}]$ . Also, the last three displays imply (5.4). ∎

Let $(N_{0},N_{1})\in\mathcal{A}$ . Let $\mathcal{I}$ and $\mathcal{R}$ be as in (4.7) and (4.8), respectively. Since we are presently only interested in $\bm{c}\in\mathcal{S}$ , we may and do assume $\lvert\mathcal{I}\rvert\geq 1$ . For each $\bm{c}\in\mathcal{S}$ , we have

\begin{split}\lvert\Diamond_{\bm{c},N_{0},N_{1}}\rvert&\leq\sum_{n_{0}\in[N_{0},2N_{0})}\lvert a^{\prime}_{\bm{c}}(n_{0})\,\heartsuit_{\bm{c},n_{0},N_{1}}\rvert\\ &\ll N^{(1-m)/2}\,\lVert I_{\bm{c}}\rVert_{1,\infty;N}\,\sum_{n_{0}\in[N_{0},2N_{0})}\lvert a^{\prime}_{\bm{c}}(n_{0})\rvert\,\int_{x\in[N_{1},2N_{1}]}\lvert B_{\bm{c}}([N_{1},x))\rvert\,d\nu(x),\end{split}

where the first and second inequality are justified by the triangle inequality and Lemma 5.2, respectively. Abbreviating $B_{\bm{c}}([N_{1},x))$ to $B_{\bm{c}}(x)$ for convenience, we deduce that

(5.5)

\sum_{\bm{c}\in\mathcal{R}:\,\Delta(\bm{c})\neq 0}\lvert\Diamond_{\bm{c},N_{0},N_{1}}\rvert\ll_{\epsilon}X^{m+\epsilon}Q_{1}\,\biggl{(}\,\sum_{\bm{c}\in\mathcal{R}:\,\Delta(\bm{c})\neq 0}\left(\int_{x\in[N_{1},2N_{1}]}\lvert B_{\bm{c}}(x)\rvert\,d\nu\right)^{\!2}\,\biggr{)}^{\!1/2}

by the Cauchy–Schwarz inequality and Lemma 4.9, where

(5.6)

Q_{1}\colonequals N^{(1-m)/2}N_{0}^{1/2+(m-\lvert\mathcal{I}\rvert)/6}(X/N)^{1-(m+\lvert\mathcal{I}\rvert)/4}\max[Z^{1+(\lvert\mathcal{I}\rvert-m)/4},1].

Now, for the rest of §5, we assume Hypothesis 2.4. We have

\left(\int_{x\in[N_{1},2N_{1}]}\lvert B_{\bm{c}}(x)\rvert\,d\nu\right)^{\!2}\ll\int_{x\in[N_{1},2N_{1}]}\lvert B_{\bm{c}}(x)\rvert^{2}\,d\nu

by the Cauchy–Schwarz inequality, so

\begin{split}\biggl{(}\,\sum_{\bm{c}\in\mathcal{R}:\,\Delta(\bm{c})\neq 0}\left(\int_{x\in[N_{1},2N_{1}]}\lvert B_{\bm{c}}(x)\rvert\,d\nu\right)^{\!2}\,\biggr{)}^{\!1/2}&\ll\biggl{(}\,\int_{x\in[N_{1},2N_{1}]}\sum_{\bm{c}\in\mathcal{R}:\,\Delta(\bm{c})\neq 0}\lvert B_{\bm{c}}(x)\rvert^{2}\,d\nu\biggr{)}^{\!1/2}\\ &\ll_{\epsilon}X^{\epsilon}Q_{2}\end{split}

by (4.15), where

(5.7)

Q_{2}\colonequals\min\left(Z^{m}N_{1},Z^{\lvert\mathcal{I}\rvert}N_{1}^{\max(2,1+(m-\lvert\mathcal{I}\rvert)/3)}\right)^{1/2}.

Lemma 5.3.

We have $Q_{1}Q_{2}\ll_{\epsilon_{0}}X^{3/2-m/4+O(\epsilon_{0})}$ .

Proof.

We split the proof into four cases.

Case 1: $\lvert\mathcal{I}\rvert=m$ . Then $Q_{2}=(Z^{m}N_{1})^{1/2}$ , since $\lvert\mathcal{I}\rvert=m$ and $N_{1}\geq 1$ . Therefore, $Q_{1}Q_{2}=Q_{3}$ , where

(5.8)

Q_{3}\colonequals Z^{m/2}N_{1}^{1/2}\cdot N^{(1-m)/2}N_{0}^{1/2}(X/N)^{1-m/2}\max[Z,1].

But $Q_{3}=Z^{m/2}X^{1-m/2}\max[Z,1]$ , since $N_{1}N_{0}=N$ . By (4.2) we have $Z=X^{1/2+\epsilon_{0}}\geq 1$ , so

Q_{3}=X^{1-m/2}Z^{1+m/2}=X^{3/2-m/4+(1+m/2)\epsilon_{0}}.

Thus $Q_{1}Q_{2}=Q_{3}\ll_{\epsilon_{0}}X^{3/2-m/4+O(\epsilon_{0})}$ .

Case 2: $\lvert\mathcal{I}\rvert=m-1$ and $N_{1}\geq Z$ . Then $Q_{2}=(Z^{m}N_{1})^{1/2}$ , by (5.7). Therefore, $Q_{1}Q_{2}=Q_{4}$ , where

Q_{4}\colonequals Z^{m/2}N^{1-m/2}N_{0}^{(m-\lvert\mathcal{I}\rvert)/6}(X/N)^{1-(m+\lvert\mathcal{I}\rvert)/4}\max[Z^{1+(\lvert\mathcal{I}\rvert-m)/4},1],

since $N_{1}N_{0}=N$ . Since $(m-\lvert\mathcal{I}\rvert)/6\geq 0$ and $N_{0}=N/N_{1}\leq N/Z$ , we have

(5.9)

Q_{4}\leq Z^{m/2}N^{1-m/2}(N/Z)^{(m-\lvert\mathcal{I}\rvert)/6}(X/N)^{1-(m+\lvert\mathcal{I}\rvert)/4}\max[Z^{1+(\lvert\mathcal{I}\rvert-m)/4},1].

The right-hand side of (5.9) is decreasing as a function of $N$ , because

(5.10)

1-m/2+(m-\lvert\mathcal{I}\rvert)/6-1+(m+\lvert\mathcal{I}\rvert)/4=(\lvert\mathcal{I}\rvert-m)/12<0.

Since $N\geq N_{1}\geq Z$ , it follows that

\begin{split}Q_{4}&\leq Z^{m/2}Z^{1-m/2}(Z/Z)^{(m-\lvert\mathcal{I}\rvert)/6}(X/Z)^{1-(m+\lvert\mathcal{I}\rvert)/4}\max[Z^{1+(\lvert\mathcal{I}\rvert-m)/4},1]\\ &\ll_{\epsilon_{0}}X^{O(\epsilon_{0})}X^{1-(m+\lvert\mathcal{I}\rvert)/8}\max[X^{1/2+(\lvert\mathcal{I}\rvert-m)/8},1],\end{split}

since $Z=X^{1/2+\epsilon_{0}}$ . But $\lvert\mathcal{I}\rvert=m-1$ , so

(5.11)

X^{1-(m+\lvert\mathcal{I}\rvert)/8}\max[X^{1/2+(\lvert\mathcal{I}\rvert-m)/8},1]=\max[X^{3/2-m/4},X^{9/8-m/4}]=X^{3/2-m/4}.

Thus $Q_{1}Q_{2}=Q_{4}\ll_{\epsilon_{0}}X^{3/2-m/4+O(\epsilon_{0})}$ .

Case 3: $1\leq\lvert\mathcal{I}\rvert\leq m-2$ . By (5.7), we have $Q_{2}\leq(Z^{\lvert\mathcal{I}\rvert}N_{1}^{\max(2,1+(m-\lvert\mathcal{I}\rvert)/3)})^{1/2}$ . Since $N_{1}N_{0}=N$ , it follows that $Q_{1}Q_{2}\leq Q_{5}$ , where

Q_{5}\colonequals Z^{\lvert\mathcal{I}\rvert/2}N_{1}^{\max(1/2,(m-\lvert\mathcal{I}\rvert)/6)}N^{1-m/2}N_{0}^{(m-\lvert\mathcal{I}\rvert)/6}(X/N)^{1-(m+\lvert\mathcal{I}\rvert)/4}\max[Z^{1+(\lvert\mathcal{I}\rvert-m)/4},1].

Since $N_{0}\geq 1$ and $N_{1}N_{0}=N$ , we have $N_{1}^{\max(1/2,(m-\lvert\mathcal{I}\rvert)/6)}N_{0}^{(m-\lvert\mathcal{I}\rvert)/6}\leq N^{\max(1/2,(m-\lvert\mathcal{I}\rvert)/6)}$ . Thus

(5.12)

Q_{5}\leq Z^{\lvert\mathcal{I}\rvert/2}N^{\max(1/2,(m-\lvert\mathcal{I}\rvert)/6)}N^{1-m/2}(X/N)^{1-(m+\lvert\mathcal{I}\rvert)/4}\max[Z^{1+(\lvert\mathcal{I}\rvert-m)/4},1].

The right-hand side of (5.12) is weakly decreasing in $N$ , because

\begin{split}\max(1/2,(m-\lvert\mathcal{I}\rvert)/6)+1-m/2-1+(m+\lvert\mathcal{I}\rvert)/4&=\max(1/2,(m-\lvert\mathcal{I}\rvert)/6)+(\lvert\mathcal{I}\rvert-m)/4\\ &\leq 0,\end{split}

in view of the inequality $\lvert\mathcal{I}\rvert-m\leq-2$ . Since $N\geq 1$ and $\lvert\mathcal{I}\rvert\leq m$ , it follows that

\begin{split}Q_{5}&\leq Z^{\lvert\mathcal{I}\rvert/2}X^{1-(m+\lvert\mathcal{I}\rvert)/4}\max[Z^{1+(\lvert\mathcal{I}\rvert-m)/4},1]\\ &\leq Z^{\lvert\mathcal{I}\rvert/2}X^{1-(m+\lvert\mathcal{I}\rvert)/4}Z\\ &\ll_{\epsilon_{0}}X^{O(\epsilon_{0})}X^{3/2-m/4},\end{split}

since $Z=X^{1/2+\epsilon_{0}}$ . Thus $Q_{1}Q_{2}\leq Q_{5}\ll_{\epsilon_{0}}X^{3/2-m/4+O(\epsilon_{0})}$ .

Case 4: $\lvert\mathcal{I}\rvert=m-1$ and $N_{1}\leq Z$ . Arguing as in Case 3, we have $Q_{1}Q_{2}\leq Q_{5}$ . But if we hold $N_{1}$ constant, and plug $N_{0}=N/N_{1}$ into $Q_{5}$ , then $Q_{5}$ is decreasing in $N$ , by (5.10). Since $N\geq N_{1}$ , it follows that $Q_{5}\leq Q_{6}$ , where

Q_{6}\colonequals Z^{\lvert\mathcal{I}\rvert/2}N_{1}^{\max(1/2,(m-\lvert\mathcal{I}\rvert)/6)}N_{1}^{1-m/2}(X/N_{1})^{1-(m+\lvert\mathcal{I}\rvert)/4}\max[Z^{1+(\lvert\mathcal{I}\rvert-m)/4},1].

But $Q_{6}$ is increasing in $N_{1}$ , because

\max(1/2,(m-\lvert\mathcal{I}\rvert)/6)+1-m/2-1+(m+\lvert\mathcal{I}\rvert)/4=1/4>0,

in view of the equality $\lvert\mathcal{I}\rvert=m-1$ . Since $N_{1}\leq Z$ and $\lvert\mathcal{I}\rvert=m-1$ , it follows that

\begin{split}Q_{6}&\leq Z^{\lvert\mathcal{I}\rvert/2}Z^{1/2}Z^{1-m/2}(X/Z)^{1-(m+\lvert\mathcal{I}\rvert)/4}\max[Z^{1+(\lvert\mathcal{I}\rvert-m)/4},1]\\ &=Z\,(X/Z)^{1-(m+\lvert\mathcal{I}\rvert)/4}\max[Z^{1+(\lvert\mathcal{I}\rvert-m)/4},1]\\ &\ll_{\epsilon_{0}}X^{O(\epsilon_{0})}X^{1-(m+\lvert\mathcal{I}\rvert)/8}\max[X^{1/2+(\lvert\mathcal{I}\rvert-m)/8},1],\end{split}

since $Z=X^{1/2+\epsilon_{0}}$ . But $\lvert\mathcal{I}\rvert=m-1$ , so it follows from (5.11) that $Q_{6}\ll_{\epsilon_{0}}X^{O(\epsilon_{0})}X^{3/2-m/4}$ . Thus $Q_{1}Q_{2}\leq Q_{5}\leq Q_{6}\ll_{\epsilon_{0}}X^{3/2-m/4+O(\epsilon_{0})}$ . ∎

Remark 5.4.

Interestingly, the quantity $Q_{3}$ in (5.8) is constant over $(N_{0},N_{1})\in\mathcal{A}$ .

By Lemma 5.3, the left-hand side of (5.5) is $\ll_{\epsilon_{0}}X^{m+O(\epsilon_{0})}X^{3/2-m/4}$ . Upon summing over $(N_{0},N_{1})\in\mathcal{A}$ and the set of $2^{m}-1$ possible sets $\mathcal{R}$ , it follows from (5.3) that

\Sigma_{1}\ll_{\epsilon_{0}}X^{-3}X^{m+O(\epsilon_{0})}X^{3/2-m/4}=X^{3(m-2)/4+O(\epsilon_{0})}.

This yields the desired inequality, (5.1).

6. Contribution from the central terms

Here we address the $\bm{c}=\bm{0}$ contribution to (4.6), using the theory of $I_{\bm{0}}(n)$ developed in [heath1996new]. We roughly follow [heath1996new]*§12, par. 2. Let

(6.1)

\Sigma_{2}\colonequals X^{-3}\sum_{n\geq 1}n^{(1-m)/2}S^{\natural}_{\bm{0}}(n)I_{\bm{0}}(n).

We begin with a slight extension of [vaughan1997hardy]*Lemma 4.9.

Lemma 6.1.

If $N\geq 1$ , then $\sum_{n\in[N,2N)}n^{-m}\lvert S_{\bm{0}}(n)\rvert\ll_{\epsilon}N^{(4-m)/3+\epsilon}$ .

Proof.

We have $S^{\natural}_{\bm{0}}(n)\ll_{\epsilon}n^{1/2+\epsilon}\operatorname{cub}(n)^{m/6}$ by Proposition 4.6. Thus

n^{-m}S_{\bm{0}}(n)\ll_{\epsilon}n^{1-m/2+\epsilon}\operatorname{cub}(n)^{m/6}.

Taking $t=m/6$ in Lemma 3.3(3), we get

\sum_{n\in[N,2N)}n^{-m}\lvert S_{\bm{0}}(n)\rvert\ll_{\epsilon}N^{1-m/2+\epsilon}\max(N,N^{1/3+m/6})=N^{(4-m)/3+\epsilon},

where we note that $\max(N,N^{1/3+m/6})=N^{1/3+m/6}$ because $N\geq 1$ and $m\geq 4$ . ∎

Lemma 6.1 implies, in particular, the familiar fact that the singular series

(6.2)

\mathfrak{S}\colonequals\sum_{n\geq 1}n^{-m}S_{\bm{0}}(n)

converges absolutely for $m\geq 5$ . It is also known that the real density

(6.3)

\sigma_{\infty,w}\colonequals\lim_{\epsilon\to 0}{(2\epsilon)^{-1}\int_{\lvert F(\bm{x})\rvert\leq\epsilon}w(\bm{x})\,d\bm{x}}

exists; see e.g. [heath1996new]*Theorem 3. Yet for all $n\ll Y$ , [heath1996new]*Lemma 13 implies

(6.4)

X^{-m}I_{\bm{0}}(n)=\sigma_{\infty,w}+O_{A}((n/Y)^{A}),

for all $A>0$ . If $m\geq 5$ , then via (6.4) with $A=(m-4)/3$ , we get

\begin{split}&\sum_{n\leq M_{0}(F,w)Y}n^{-m}S_{\bm{0}}(n)X^{-m}I_{\bm{0}}(n)\\ &=\sigma_{\infty,w}\sum_{n\leq M_{0}(F,w)Y}n^{-m}S_{\bm{0}}(n)+\sum_{n\leq M_{0}(F,w)Y}O((n/Y)^{(m-4)/3}n^{-m}\lvert S_{\bm{0}}(n)\rvert)\\ &=\sigma_{\infty,w}\mathfrak{S}+O_{\epsilon}(Y^{(4-m)/3+\epsilon}),\end{split}

by Lemma 6.1 and (6.2). Also, by Proposition 4.1, we have $I_{\bm{0}}(n)=0$ for all $n>M_{0}(F,w)Y$ . Since $n^{-m}S_{\bm{0}}(n)=n^{(1-m)/2}S^{\natural}_{\bm{0}}(n)$ and $Y=X^{3/2}$ , it follows that if $m\geq 5$ , then

(6.5)

\Sigma_{2}=X^{m-3}\,[\sigma_{\infty,w}\mathfrak{S}+O_{\epsilon}(X^{(4-m)/2+\epsilon})]=\sigma_{\infty,w}\mathfrak{S}X^{m-3}+O_{\epsilon}(X^{(m-2)/2+\epsilon}),

where $\Sigma_{2}$ is the quantity defined in (6.1). On the other hand, for all $m\geq 4$ ,

(6.6)

\Sigma_{2}\ll X^{m-3}\sum_{n\leq M_{0}(F,w)Y}n^{-m}\lvert S_{\bm{0}}(n)\rvert\ll_{\epsilon}X^{m-3+\epsilon}

by Proposition 4.1 and Lemma 6.1, since $I_{\bm{0}}(n)\ll X^{m}$ by [heath1996new]*Lemma 16.

7. Contribution from singular hyperplane sections

In this section, we study the quantity

(7.1)

\Sigma_{3}\colonequals X^{-3}\sum_{n\geq 1}\,\sum_{\bm{c}\in[-Z,Z]^{m}:\,\Delta(\bm{c})=0,\;\bm{c}\neq\bm{0}}n^{(1-m)/2}S^{\natural}_{\bm{c}}(n)I_{\bm{c}}(n).

We will prove the following result, extending work of Heath-Brown. Recall the definitions of $N_{F,w}(X)$ and $N^{\prime}_{F,w}(X)$ from (2.14) and (2.15), respectively.

Theorem 7.1.

If $m\geq 5$ , then

(7.2)

\Sigma_{3}\ll_{\epsilon_{0}}X^{3(m-2)/4+O(\epsilon_{0})}.

If $m=4$ , then

(7.3)

\Sigma_{3}=N_{F,w}(X)-N^{\prime}_{F,w}(X)+O_{\epsilon_{0}}(X^{3(m-2)/4+O(\epsilon_{0})}).

The cases $m=4$ and $m=6$ of this result are due to Heath-Brown. For instance, the estimate (7.3) for $m=4$ follows directly from [heath1998circle]*Lemmas 7.2 and 8.1, in view of the tail estimate (4.3). Therefore, we may and do assume $m\geq 5$ , for the rest of §7.

We combine ideas from [hooley1986HasseWeil] and [heath1998circle]. Let $\mathcal{I}$ and $\mathcal{R}$ be as in (4.7) and (4.8), respectively. Since we are only interested in $\bm{c}\neq\bm{0}$ , we may and do assume $\lvert\mathcal{I}\rvert\geq 1$ . Let $\mathcal{C}$ and $C$ be as in (4.12), for some $C_{i}\in\{2^{t}:t\in\mathbb{Z}_{\geq 0}\}$ with $1\leq C_{i}\leq Z$ .

By Proposition 4.1, the sum $\Sigma_{3}$ from (7.1) is supported on $n\leq M_{0}(F,w)Y$ . Let

(7.4)

\Sigma_{4}\colonequals X^{-3}\sum_{n\leq M_{0}(F,w)Y}\sum_{\bm{c}\in\mathcal{C}:\,\Delta(\bm{c})=0}n^{(1-m)/2}S^{\natural}_{\bm{c}}(n)I_{\bm{c}}(n).

Now consider an element $\bm{c}\in\mathcal{C}$ with $\Delta(\bm{c})=0$ , assuming such a $\bm{c}$ exists. Denote the nonempty fibers of the map $\mathcal{I}\to\mathbb{Q}^{\times}/(\mathbb{Q}^{\times})^{2},\;i\mapsto F_{i}c_{i}\bmod{(\mathbb{Q}^{\times})^{2}}$ by

\mathcal{I}(k)\colonequals\{i\in\mathcal{I}:F_{i}c_{i}\equiv g_{k}\bmod{(\mathbb{Q}^{\times})^{2}}\},

for $1\leq k\leq K$ , say, where the $g_{k}$ are signed, nonzero square-free integers. Trivially, we have $\sum_{1\leq k\leq K}\lvert\mathcal{I}(k)\rvert=\lvert\mathcal{I}\rvert$ . For each $i\in\mathcal{I}(k)$ , we may write

(7.5)

c_{i}=g_{k}F_{i}^{-1}e_{i}^{2}

with $e_{i}\in\mathbb{Z}$ . Moreover, by (2.1) and the $\mathbb{Q}$ -linear independence of square roots of distinct square-free integers, we may choose the signs of the integers $e_{i}$ so that

(7.6)

\sum_{i\in\mathcal{I}(k)}F_{i}(e_{i}/F_{i})^{3}=0.

Since $c_{i}\neq 0$ implies $e_{i}\neq 0$ for all $i\in\mathcal{I}(k)$ , we immediately deduce from (7.6) that

(7.7)

\lvert\mathcal{I}(k)\rvert\geq 2.

We now prove a general lemma that will allow us, in Lemma 7.3, to exploit the structure uncovered in the previous paragraph.

Lemma 7.2.

Let $J\in\mathbb{Z}_{\geq 2}$ , let $d_{1},\dots,d_{J}\in\mathbb{Z}_{\geq 1}$ , and let $G,E_{1},\dots,E_{J}\in\mathbb{R}_{>0}$ . Then

\sum_{\begin{subarray}{c}1\leq g\leq G:\\ \mu(g)^{2}=1\end{subarray}}\,\prod_{1\leq i\leq J}\sum_{\begin{subarray}{c}1\leq e_{i}\leq E_{i}:\\ d_{i}\mid\operatorname{sq}(ge_{i}^{2})\end{subarray}}d_{i}^{1/2}\leq\prod_{1\leq i\leq J}(2^{\omega(d_{i})}G^{1/2}E_{i}).

Proof.

By Hölder’s inequality over $g$ , we may assume that $E_{1}=\dots=E_{J}=E$ and $d_{1}=\dots=d_{J}=d$ , say. Let $S\colonequals\{h\mid d:\mu(h)^{2}=1\}$ . Now consider integers $g,e\geq 1$ with $g$ square-free. Then $\operatorname{sq}(ge^{2})=\gcd(g,e)e^{2}$ . Therefore, if $d\mid\operatorname{sq}(ge^{2})$ , and we let $h\colonequals\gcd(g,e,d)$ , then

h\in S,\qquad(d/h)\mid e^{2},

whence $e$ is divisible by the integer $\prod_{p\mid(d/h)}p^{\lceil v_{p}(d/h)/2\rceil}\geq(d/h)^{1/2}$ . Thus, given $h\in S$ , the number of possible $e\in[1,E]$ is at most $E/(d/h)^{1/2}$ . It follows that

(7.8)

\sum_{\begin{subarray}{c}1\leq e\leq E:\\ d\mid\operatorname{sq}(ge^{2})\end{subarray}}d^{1/2}\leq\sum_{h\in S}(d^{1/2}\cdot\bm{1}_{h\mid g}\cdot E/(d/h)^{1/2})=\sum_{h\in S}(\bm{1}_{h\mid g}\cdot h^{1/2}E),

for every square-free $g\geq 1$ . By (7.8), and Hölder’s inequality over $h$ , we get

\begin{split}\sum_{\begin{subarray}{c}1\leq g\leq G:\\ \mu(g)^{2}=1\end{subarray}}\,\biggl{(}\,\sum_{\begin{subarray}{c}1\leq e\leq E:\\ d\mid\operatorname{sq}(ge^{2})\end{subarray}}d^{1/2}\biggr{)}^{\!J}&\leq\sum_{\begin{subarray}{c}1\leq g\leq G:\\ \mu(g)^{2}=1\end{subarray}}\,\biggl{(}\,\sum_{h\in S}(\bm{1}_{h\mid g}\cdot h^{1/2}E)\biggr{)}^{\!J}\\ &\leq\sum_{\begin{subarray}{c}1\leq g\leq G:\\ \mu(g)^{2}=1\end{subarray}}\,\lvert S\rvert^{J-1}\sum_{h\in S}(\bm{1}_{h\mid g}\cdot h^{1/2}E)^{J}\\ &\leq\lvert S\rvert^{J-1}\sum_{\begin{subarray}{c}h\in S:\\ h\leq G\end{subarray}}(G/h)(h^{1/2}E)^{J}\\ &\leq\lvert S\rvert^{J}G^{J/2}E^{J},\end{split}

where in the last step we note that $h^{J/2-1}\leq G^{J/2-1}$ . This suffices, since $\lvert S\rvert=2^{\omega(d)}$ . ∎

Lemma 7.3.

Let $n\geq 1$ be an integer. Then

\sum_{\bm{c}\in\mathcal{C}:\,\Delta(\bm{c})\neq 0}n^{-1}\lvert S^{\natural}_{\bm{c}}(n)\rvert^{2}\ll_{\epsilon}n^{\epsilon}\operatorname{cub}(n)^{(m-\lvert\mathcal{I}\rvert)/3}\prod_{i\in\mathcal{I}}C_{i}^{1/2+\epsilon}.

Proof.

Let $n_{3}\colonequals\operatorname{cub}(n)$ . Fix a set $\mathcal{J}\subseteq\mathcal{I}$ with $\lvert\mathcal{J}\rvert\geq 2$ . Let $G\in\{2^{t}:t\in\mathbb{Z}_{\geq 0}\}$ , and let $E_{i}\colonequals(2F_{i}C_{i}/G)^{1/2}$ for each $i\in\mathcal{J}$ . Let $\tau(\cdot)$ be the divisor function. Then

\begin{split}&\sum_{\begin{subarray}{c}\lvert g\rvert\in[G,2G):\\ \mu(\lvert g\rvert)^{2}=1\end{subarray}}\,\prod_{i\in\mathcal{J}}\sum_{\begin{subarray}{c}\lvert e_{i}\rvert\leq(2F_{i}C_{i}/\lvert g\rvert)^{1/2}:\\ gF_{i}^{-1}e_{i}^{2}\in\mathbb{Z}\setminus\{0\}\end{subarray}}\gcd(n_{3},\operatorname{sq}(gF_{i}^{-1}e_{i}^{2}))^{1/2}\\ &\leq 2^{1+\lvert\mathcal{J}\rvert}\sum_{\begin{subarray}{c}g\in[G,2G):\\ \mu(g)^{2}=1\end{subarray}}\,\prod_{i\in\mathcal{J}}\sum_{1\leq e_{i}\leq E_{i}}\gcd(n_{3},\operatorname{sq}(ge_{i}^{2}))^{1/2}\\ &\leq 2^{1+\lvert\mathcal{J}\rvert}\sum_{\begin{subarray}{c}g\in[G,2G):\\ \mu(g)^{2}=1\end{subarray}}\,\prod_{i\in\mathcal{J}}\sum_{d_{i}\mid n_{3}}\sum_{\begin{subarray}{c}1\leq e_{i}\leq E_{i}:\\ d_{i}\mid\operatorname{sq}(ge_{i}^{2})\end{subarray}}d_{i}^{1/2}\\ &\leq 2^{1+\lvert\mathcal{J}\rvert}\prod_{i\in\mathcal{J}}(\tau(n_{3})^{2}G^{1/2}E_{i})\\ &=2^{1+\lvert\mathcal{J}\rvert}\tau(n_{3})^{2\lvert\mathcal{J}\rvert}\prod_{i\in\mathcal{J}}(2F_{i}C_{i})^{1/2},\end{split}

where in the penultimate step we use Lemma 7.2 for each possible choice of divisors $d_{i}\mid n_{3}$ , and we note that $2^{\omega(d_{i})}\leq 2^{\omega(n_{3})}\leq\tau(n_{3})$ . Moreover, if $G>\min_{i\in\mathcal{J}}(2F_{i}C_{i})$ , then

\sum_{\begin{subarray}{c}\lvert g\rvert\in[G,2G):\\ \mu(\lvert g\rvert)^{2}=1\end{subarray}}\,\prod_{i\in\mathcal{J}}\sum_{\begin{subarray}{c}\lvert e_{i}\rvert\leq(2F_{i}C_{i}/\lvert g\rvert)^{1/2}:\\ gF_{i}^{-1}e_{i}^{2}\in\mathbb{Z}\setminus\{0\}\end{subarray}}\gcd(n_{3},\operatorname{sq}(gF_{i}^{-1}e_{i}^{2}))^{1/2}=0,

since the sum over one of the variables $e_{i}\in\mathbb{Z}\setminus\{0\}$ is empty. On summing the penultimate display over all $G\in\{2^{t}:t\in\mathbb{Z}_{\geq 0}\}$ with $G\leq\min_{i\in\mathcal{J}}(2F_{i}C_{i})$ , we conclude that

(7.9)

\sum_{\begin{subarray}{c}g\in\mathbb{Z}\setminus\{0\}:\\ \mu(\lvert g\rvert)^{2}=1\end{subarray}}\,\prod_{i\in\mathcal{J}}\sum_{\begin{subarray}{c}\lvert e_{i}\rvert\leq(2F_{i}C_{i}/\lvert g\rvert)^{1/2}:\\ gF_{i}^{-1}e_{i}^{2}\in\mathbb{Z}\setminus\{0\}\end{subarray}}\gcd(n_{3},\operatorname{sq}(gF_{i}^{-1}e_{i}^{2}))^{1/2}\ll_{\lvert\mathcal{J}\rvert,\epsilon}n_{3}^{\epsilon}\prod_{i\in\mathcal{J}}(2F_{i}C_{i})^{1/2+\epsilon}.

Recall the constraints (7.5) and (7.7) on $\{\bm{c}\in\mathcal{C}:\Delta(\bm{c})=0\}$ . Applying (7.9) with $\mathcal{J}=\mathcal{I}(k)$ , for each $k\in[1,K]$ , and multiplying the resulting $K$ inequalities, we get

(7.10)

\prod_{1\leq k\leq K}\sum_{\begin{subarray}{c}g_{k}\in\mathbb{Z}\setminus\{0\}:\\ \mu(\lvert g_{k}\rvert)^{2}=1\end{subarray}}\,\prod_{i\in\mathcal{I}(k)}\sum_{\begin{subarray}{c}\lvert e_{i}\rvert\leq(2F_{i}C_{i}/\lvert g_{k}\rvert)^{1/2}:\\ g_{k}F_{i}^{-1}e_{i}^{2}\in\mathbb{Z}\setminus\{0\}\end{subarray}}\,\gcd(n_{3},\operatorname{sq}(g_{k}F_{i}^{-1}e_{i}^{2}))^{1/2}\\ \ll_{\epsilon}n_{3}^{\epsilon}\prod_{i\in\mathcal{I}}C_{i}^{1/2+\epsilon},

since $K\leq\lvert\mathcal{I}\rvert\leq m$ , and the variables $m,F_{i}$ are fixed. On summing (7.10) over all possible choices for the sets $\mathcal{I}(k)\subseteq\mathcal{I}$ , we deduce that

(7.11)

\sum_{\bm{c}\in\mathcal{C}:\,\Delta(\bm{c})=0}\prod_{i\in\mathcal{I}}\gcd(n_{3},\operatorname{sq}(c_{i}))^{1/2}\ll_{\epsilon}n_{3}^{\epsilon}\prod_{i\in\mathcal{I}}C_{i}^{1/2+\epsilon}.

Lemma 7.3 follows immediately from (4.9) and (7.11). ∎

Remark 7.4.

Interestingly, the proof of (7.11) uses the constraint (7.6) only through (7.7).

Taking $n_{3}=1$ in (7.11) implies

\lvert\{\bm{c}\in\mathcal{C}:\Delta(\bm{c})=0\}\rvert\ll_{\epsilon}\prod_{i\in\mathcal{I}}C_{i}^{1/2+\epsilon}.

Therefore, Lemma 7.3 implies

(7.12)

\sum_{\bm{c}\in\mathcal{C}:\,\Delta(\bm{c})=0}n^{-1/2}\lvert S^{\natural}_{\bm{c}}(n)\rvert\ll_{\epsilon}n^{\epsilon}\operatorname{cub}(n)^{(m-\lvert\mathcal{I}\rvert)/6}\prod_{i\in\mathcal{I}}C_{i}^{1/2+\epsilon},

by the Cauchy–Schwarz inequality over $\bm{c}$ .

Let $N\in\{2^{t}:t\in\mathbb{Z}_{\geq 0}\}$ with $1\leq N\leq M_{0}(F,w)Y$ . By Lemma 4.7, (7.12), and the $t=(m-\lvert\mathcal{I}\rvert)/6$ case of Lemma 3.3(3), the sum

\Sigma_{5}\colonequals X^{-3}\sum_{n\in[N,2N)}\sum_{\bm{c}\in\mathcal{C}:\,\Delta(\bm{c})=0}n^{(1-m)/2}\lvert S^{\natural}_{\bm{c}}(n)I_{\bm{c}}(n)\rvert

satisfies the bound $\Sigma_{5}\ll_{\epsilon}X^{m-3+\epsilon}Q_{7}$ , where

\begin{split}Q_{7}&\colonequals N^{1-m/2}(XC/N)^{1-(m+\lvert\mathcal{I}\rvert)/4}\max(N,N^{1/3+(m-\lvert\mathcal{I}\rvert)/6})\,C^{\lvert\mathcal{I}\rvert/2}\\ &=X^{1-(\lvert\mathcal{I}\rvert+m)/4}\max(N^{1+(\lvert\mathcal{I}\rvert-m)/4},N^{1/3+(\lvert\mathcal{I}\rvert-m)/12})\,C^{1+(\lvert\mathcal{I}\rvert-m)/4}.\end{split}

Since $N^{1+(\lvert\mathcal{I}\rvert-m)/4}=(N^{1/3+(\lvert\mathcal{I}\rvert-m)/12})^{3}$ , we will analyze $Q_{7}$ according to the sign of

\mathfrak{e}\colonequals 1+(\lvert\mathcal{I}\rvert-m)/4.

Case 1: $\mathfrak{e}\leq 0$ . Then, since $N,C\gg 1$ , we have

Q_{7}\ll X^{1-(\lvert\mathcal{I}\rvert+m)/4}\leq X^{(6-m)/4},

where the final inequality holds because $X\geq 1$ and $\lvert\mathcal{I}\rvert\geq-2$ .

Case 2: $\mathfrak{e}\geq 0$ . Then, since $N\ll Y$ and $C\ll Z$ , we have

Q_{7}\ll X^{1-(\lvert\mathcal{I}\rvert+m)/4}Y^{1+(\lvert\mathcal{I}\rvert-m)/4}Z^{1+(\lvert\mathcal{I}\rvert-m)/4}.

Plugging in (4.1) and (4.2), we get

Q_{7}\ll_{\epsilon_{0}}X^{1-(\lvert\mathcal{I}\rvert+m)/4+O(\epsilon_{0})}(X^{2})^{1+(\lvert\mathcal{I}\rvert-m)/4}=X^{3+(\lvert\mathcal{I}\rvert-3m)/4+O(\epsilon_{0})}.

Moreover, if $\lvert\mathcal{I}\rvert\leq 2m-6$ , then $3+(\lvert\mathcal{I}\rvert-3m)/4\leq(6-m)/4$ .

If $m\geq 6$ , then $1\leq\lvert\mathcal{I}\rvert\leq m\leq 2m-6$ , so regardless of what $\lvert\mathcal{I}\rvert$ is, it follows that

\begin{split}\Sigma_{5}&\ll_{\epsilon_{0}}X^{m-3+\epsilon_{0}}Q_{7}\\ &\ll_{\epsilon_{0}}X^{m-3+\epsilon_{0}}X^{(6-m)/4+O(\epsilon_{0})}\\ &=X^{3(m-2)/4+O(\epsilon_{0})},\end{split}

whence by summing over all possibilities for $N$ and $\mathcal{C}$ we get

\Sigma_{3},\Sigma_{4}\ll_{\epsilon_{0}}X^{3(m-2)/4+O(\epsilon_{0})},

where $\Sigma_{3},\Sigma_{4}$ are as defined in (7.1) and (7.4), respectively. This completes the proof of (7.2) for $m\geq 6$ . For the rest of §7, we relinquish the previous definitions of $\mathcal{C}$ and $C$ .

For $m=5$ , we first show that a natural extension of [heath1998circle]*Lemma 7.1 holds.

Lemma 7.5.

If $5\leq m\leq 6$ and $C\gg 1$ , then $\lvert\{\bm{c}\in\mathbb{Z}^{m}\cap[-C,C]^{m}:\Delta(\bm{c})=0\}\rvert\ll_{\epsilon}C^{m-3+\epsilon}$ .

Proof.

For $m=6$ , this follows directly from [heath1998circle]*Lemma 7.1. Now suppose $m=5$ . A partition of $m$ is an infinite, weakly decreasing sequence of nonnegative integers $\lambda_{1},\lambda_{2},\dots$ , such that $\sum_{k\geq 1}\lambda_{k}=m$ . For any partition of $m$ , let

e_{k}\colonequals 2\cdot\bm{1}_{2\leq\lambda_{k}\leq 4}+(\lambda_{k}-2)\cdot\bm{1}_{\lambda_{k}\geq 5}

for $k\geq 1$ . Let $\theta$ denote the maximum value of $\frac{1}{2}\sum_{k\geq 1}e_{k}$ over all partitions of $m$ . By [heath1998circle]*p. 687, we have $\lvert\{\bm{c}\in\mathbb{Z}^{m}\cap[-C,C]^{m}:\Delta(\bm{c})=0\}\rvert\ll_{\epsilon}C^{\theta+\epsilon}$ .

Clearly $\lambda_{3}\leq\lfloor m/3\rfloor=1$ , so $e_{k}=0$ for all $k\geq 3$ . If $\lambda_{2}\leq 1$ , then $e_{k}=0$ for all $k\geq 2$ , so $\sum_{k\geq 1}e_{k}=e_{1}\leq m-2$ . If $\lambda_{2}\geq 2$ , then $\lambda_{1}\leq m-\lambda_{2}\leq 3$ , so $e_{k}\leq 2$ for all $k\geq 1$ , whence $\sum_{k\geq 1}e_{k}=e_{1}+e_{2}\leq 4$ . In either case, $\sum_{k\geq 1}e_{k}\leq 4$ . Therefore, $\theta\leq 2=m-3$ . ∎

We now recall a bound from [heath1998circle] that is valid for all $m\geq 4$ .

Lemma 7.6.

Fix $\varepsilon>0$ . Suppose $1\ll N\ll X^{3/2}$ and $1\ll C\ll X^{1/2+\varepsilon}$ . Let

A=\sum_{N<q\leq 2N}\,\sum_{C<\lVert\bm{c}\rVert\leq 2C:\,\Delta(\bm{c})=0}\,q^{-m}S_{\bm{c}}(q)I_{\bm{c}}(q).

Then there exist reals $X_{1},X_{2},X_{3}\gg 1$ and an integer $H\geq 1$ such that $X_{1}X_{2}X_{3}\asymp N$ and

A\ll_{\varepsilon}X^{m+4\varepsilon}N^{-m}X_{1}^{1+m/2}X_{2}^{2/3+2m/3}X_{3}^{1+2m/3}H^{1/2}\left(\frac{N}{XC}\right)^{\!(m-2)/4}\mathcal{N}_{1}\mathcal{N}_{2}(H),

where in terms of the quantity $\mathfrak{D}=3(\prod_{1\leq i\leq m}F_{i})^{2^{m-2}}$ from §2, we let

\begin{split}\mathcal{N}_{1}&\colonequals\sum_{(q_{1},q_{2},q_{3}):\,X_{i}<q_{i}\leq 2X_{i}}\bm{1}_{\operatorname{cub}(q_{1})=1}\bm{1}_{q_{2}=\operatorname{cub}(q_{2})}\bm{1}_{q_{3}\mid\mathfrak{D}^{\infty}},\\ \mathcal{N}_{2}(H)&\colonequals\sum_{C<\lVert\bm{c}\rVert\leq 2C}\bm{1}_{H\mid\bm{c}}\bm{1}_{\Delta(\bm{c})=0}.\end{split}

Proof.

This is immediate from [heath1998circle]*pp. 688–689, from the definition of $A$ on p. 688 to the definition of $\mathcal{N}_{2}(H)$ on p. 689. What Heath-Brown calls $P$ (resp. $X$ ), we call $X$ (resp. $N$ ). Moreover, in terms of Heath-Brown’s notation $n$ and $G$ , our $m$ and $\Delta$ satisfy $m=n$ and $\Delta(\bm{c})=3G(\bm{c})$ . However, our $C,q,\bm{c},X_{1},X_{2},X_{3},H$ match Heath-Brown’s notation. ∎

Applying Lemma 3.1 to $q_{2}$ and Lemma 3.2 to $q_{3}$ , it is clear that

\mathcal{N}_{1}\ll_{\varepsilon}X_{1}X_{2}^{1/3}X_{3}^{\varepsilon}.

Now assume $5\leq m\leq 6$ . Then $\mathcal{N}_{2}(H)=0$ unless $H\leq 2C$ , in which case

\mathcal{N}_{2}(H)\ll_{\varepsilon}(C/H)^{m-3+\varepsilon}

by Lemma 7.5. Plugging the last two displays into Lemma 7.6, with $\varepsilon\colonequals\epsilon_{0}$ , we get

A\ll_{\epsilon_{0}}X^{m+O(\epsilon_{0})}N^{-m}X_{1}^{2+m/2}X_{2}^{1+2m/3}X_{3}^{1+2m/3}H^{1/2}\left(\frac{N}{XC}\right)^{\!(m-2)/4}\left(\frac{C}{H}\right)^{\!m-3}.

Since $m-3\geq 1/2$ , we have $H^{1/2}(C/H)^{m-3}\leq C^{m-3}$ . Moreover, $m\leq 6$ implies $2+m/2\geq 1+2m/3$ , so $X_{1}^{2+m/2}X_{2}^{1+2m/3}X_{3}^{1+2m/3}\ll(X_{1}X_{2}X_{3})^{2+m/2}\asymp N^{2+m/2}$ . Thus

(7.13)

A\ll_{\epsilon_{0}}X^{m+O(\epsilon_{0})}N^{2-m/2}\left(\frac{N}{XC}\right)^{\!(m-2)/4}C^{m-3}.

Since $2-m/2+(m-2)/4=(6-m)/4\geq 0$ (resp. since $m-3\geq(m-2)/4$ ), the right-hand side of (7.13) is weakly increasing in $N$ (resp. in $C$ ). Therefore

A\ll_{\epsilon_{0}}X^{m+O(\epsilon_{0})}(X^{3/2})^{2-m/2}(X^{1/2})^{m-3}=X^{3m/4+3/2+O(\epsilon_{0})}.

Summing over $1\ll N=M_{0}(F,w)Y/2^{k_{1}}$ and $1\ll C=Z/2^{k_{2}}$ with $k_{1},k_{2}\in\mathbb{Z}_{\geq 1}$ , we get

\Sigma_{3}\ll X^{-3}X^{3m/4+3/2+O(\epsilon_{0})}=X^{3(m-2)/4+O(\epsilon_{0})},

where $\Sigma_{3}$ is the quantity defined in (7.1). This completes the proof of (7.2).

8. Proof of main results

In this section, we first prove Theorem 2.7, because it builds directly on our work in §§4–7 on the delta method. We then prove Theorem 2.3 using (2.6), (2.7), and Proposition 4.3. Finally, we combine Theorems 2.3 and 2.7 to prove Theorem 1.1.

Proof of Theorem 2.7.

By Proposition 3.4, we see that Hypothesis 2.6 implies Hypothesis 2.4. Therefore, we may and do assume Hypothesis 2.4. Now recall the quantity $\Sigma_{0}$ from (4.6). By (4.5) and the tail estimate (4.3), we have

N_{F,w}(X)-\Sigma_{0}\ll_{A,\epsilon_{0}}X^{-A}.

Case 1: $m=4$ . Then adding (5.1), (6.6), and (7.3) together, we get

\Sigma_{0}=\Sigma_{1}+\Sigma_{2}+\Sigma_{3}=N_{F,w}(X)-N^{\prime}_{F,w}(X)+O_{\epsilon_{0}}(X^{3(m-2)/4+O(\epsilon_{0})})+O_{\epsilon_{0}}(X^{m-3+\epsilon_{0}}).

It follows that $N^{\prime}_{F,w}(X)\ll_{\epsilon_{0}}X^{3(m-2)/4+O(\epsilon_{0})}$ . Let $\mathfrak{c}(F,w)\colonequals 0$ .

Case 2: $m\geq 5$ . Then adding (5.1), (6.5), and (7.2) together, we get

\Sigma_{0}=\Sigma_{1}+\Sigma_{2}+\Sigma_{3}=\mathfrak{c}(F,w)X^{m-3}+O_{\epsilon_{0}}(X^{3(m-2)/4+O(\epsilon_{0})})+O_{\epsilon_{0}}(X^{(m-2)/2+\epsilon_{0}}),

where $\mathfrak{c}(F,w)\colonequals\sigma_{\infty,w}\mathfrak{S}$ . It follows that $N_{F,w}(X)-\mathfrak{c}(F,w)X^{m-3}\ll_{\epsilon_{0}}X^{3(m-2)/4+O(\epsilon_{0})}$ .

In each case, taking $\epsilon_{0}\to 0$ gives the desired result, (2.16). ∎

Proof of Theorem 2.3.

Let $\bm{c}\in\mathcal{S}$ . Since $\Psi(\bm{c},s)$ has an Euler product, condition (1) in Definition 2.2 clearly holds. It remains to prove that conditions (2) and (3) hold.

Case 1: $\Psi(\bm{c},s)=\Phi(\bm{c},s)$ . Then conditions (2) and (3) are trivial, since

(b_{\bm{c}}(n),a^{\prime}_{\bm{c}}(n))=(S^{\natural}_{\bm{c}}(n),\bm{1}_{n=1}).

Case 2: $\Psi(\bm{c},s)=\prod_{p\nmid\Delta(\bm{c})}\Phi_{p}(\bm{c},s)$ . Then conditions (2) and (3) are trivial, since

(b_{\bm{c}}(n),a^{\prime}_{\bm{c}}(n))=(S^{\natural}_{\bm{c}}(n)\cdot\bm{1}_{\gcd(n,\Delta(\bm{c}))=1},S^{\natural}_{\bm{c}}(n)\cdot\bm{1}_{n\mid\Delta(\bm{c})^{\infty}}).

Case 3: $\Psi(\bm{c},s)\in\{\prod_{p\nmid\Delta(\bm{c})}L_{p}(s,\bm{c})^{(-1)^{m-3}},L(s,\bm{c})^{(-1)^{m-3}}\}$ . Then by (2.6), we have

(8.1)

b_{\bm{c}}(n),a_{\bm{c}}(n)\ll_{\epsilon}n^{\epsilon}.

But $a^{\prime}_{\bm{c}}=S^{\natural}_{\bm{c}}\ast a_{\bm{c}}$ , by (2.9). Therefore, condition (2) holds. Furthermore, if $p\nmid\Delta(\bm{c})$ , then $a_{\bm{c}}(p)=-b_{\bm{c}}(p)$ by (2.9) and $b_{\bm{c}}(p)=(-1)^{m-3}\lambda_{\bm{c}}(p)=E^{\natural}_{\bm{c}}(p)$ by (2.7), so

a^{\prime}_{\bm{c}}(p)=S^{\natural}_{\bm{c}}(p)+a_{\bm{c}}(p)=S^{\natural}_{\bm{c}}(p)-E^{\natural}_{\bm{c}}(p)\ll p^{-1/2}

by Proposition 4.3. Therefore, condition (3) also holds. ∎

Proof of Theorem 1.1.

Let $\Psi\colonequals L(s,\bm{c})^{(-1)^{m-3}}$ . Then $\Psi$ is an approximation of $\Phi$ , by Theorem 2.3. Moreover, $\Psi$ is standard by (8.1) and Definition 2.5. Now let $\vartheta\colonequals 1$ . Then $\gamma_{\bm{c}}(n)=\mu(n)^{m}\lambda_{\bm{c}}(n)$ by (2.11), since for all primes $p$ we have $a_{\bm{c}}(p)=(-1)^{m-2}\lambda_{\bm{c}}(p)$ by the definition of $a_{\bm{c}}$ . Upon plugging in $\mu(n)^{m}v_{n}$ for $v_{n}$ in Hypothesis 2.1, we immediately find that Hypothesis 2.6 holds. Let $\varsigma\colon\mathbb{R}\to\mathbb{R}$ be a nonnegative, smooth, compactly supported function such that $\varsigma(t)=1$ for all $t\in[1,4]$ , and $\varsigma(t)=0$ for all $t\notin[\frac{1}{2},8]$ . Let

w(\bm{x})\colonequals\varsigma({\textstyle\sum_{1\leq i\leq m}x_{i}^{2}}).

Then Theorem 2.7 implies $N_{F,w}(X)\ll_{\epsilon}X^{3(m-2)/4+\epsilon}$ for all $X\geq 1$ . Since $w(\bm{x}/2^{k})=1$ for all $\bm{x}\in\mathbb{Z}^{m}$ in the annulus $4^{k}\leq\sum_{1\leq i\leq m}x_{i}^{2}\leq 4^{k+1}$ , it follows that

\begin{split}N_{F}(X)-1&=\lvert\{\bm{x}\in\mathbb{Z}^{m}\cap[-X,X]^{m}:F(\bm{x})=0,\;\bm{x}\neq\bm{0}\}\rvert\\ &\leq\sum_{0\leq k\leq\log_{4}(4mX^{2})}N_{F,w}(2^{k})\\ &\ll_{\epsilon}\sum_{0\leq k\leq\log_{4}(4mX^{2})}(2^{k})^{3(m-2)/4+\epsilon}\\ &\ll((4mX^{2})^{1/2})^{3(m-2)/4+\epsilon}\\ &\ll_{\epsilon}X^{3(m-2)/4+\epsilon},\end{split}

for all $X\geq 1$ . This implies Theorem 1.1. ∎

Acknowledgements

I thank Peter Sarnak for suggesting projects that ultimately led to the present paper.^†^†This work was partially supported by NSF grant DMS-1802211, and the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 101034413. I also thank him for many encouraging discussions, helpful comments, and references. Thanks also to Tim Browning, Trevor Wooley, and Nina Zubrilina for helpful comments, and to Levent Alpöge and Will Sawin for some interesting old discussions. I thank Yang Liu, Evan O’Dorney, Ashwin Sah, and Mark Sellke for conversations illuminating the combinatorics of an older, counting version of the present Lemma 4.9. Finally, special thanks are due to the editors and referees for their patience and help with the exposition.

Diagonal cubic forms and the large sieve

Abstract.

Key words and phrases:

1991 Mathematics Subject Classification:

1. Introduction

Theorem 1.1.

Conventions

2. Framework and results

Hypothesis 2.1.

Proof framework

Definition 2.2.

Theorem 2.3.

Hypotheses

Hypothesis 2.4.

Definition 2.5.

Hypothesis 2.6.

Results

Theorem 2.7.

3. A conversion between standard coefficients

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Proposition 3.4.

Proof.

4. Delta method ingredients

Proposition 4.1 ([heath1996new]*par. 1 of §7).

Lemma 4.2 ([heath1998circle]*(3.9)).

Proposition 4.3.

Proof.

Proposition 4.4.

Proof.

Proposition 4.5.

Proof.

Proposition 4.6.

Proof.

Lemma 4.7 ([heath1996new, heath1998circle]).

Proof.

Proposition 4.8.

Proof.

Lemma 4.9.

Proof.

Proposition 4.10.

Proof.

Lemma 4.11.

Proof.

Lemma 4.12.

Proof.

5. Contribution from smooth hyperplane sections

Theorem 5.1.

Lemma 5.2.

Proof.

Lemma 5.3.

Proof.

Remark 5.4.

6. Contribution from the central terms

Lemma 6.1.

Proof.

7. Contribution from singular hyperplane sections

Theorem 7.1.

Lemma 7.2.

Proof.

Lemma 7.3.

Proof.

Remark 7.4.

Lemma 7.5.

Proof.

Lemma 7.6.

Proof.

8. Proof of main results

Proof of Theorem 2.7.

Proof of Theorem 2.3.

Proof of Theorem 1.1.

Acknowledgements

References