1. Introduction
The Erdős-Hooley
$\Delta$
-function is defined by the formulae
\[\Delta(n,u)\,{:\!=}\,|\{d\,:\,d\mid n,\,{{\rm e}}^u\lt d\leqslant {{\rm e}}^{u+1}\}|,\quad \Delta(n)\,{:\!=}\,\max_{u\in{\mathbb R}}\Delta(n,u)\qquad (n\in {{\mathbb N}}^*).\]
Recent, spectacular progress on the mean-value of the
$\Delta$
-function has been obtained in [
Reference Ford, Koukoulopoulos and Tao9
,
Reference Koukoulopoulos and Tao17
], and refined in [
Reference de la Bretèche and Tenenbaum4
]. This invites one to revisit some of the applications linked to this quantity. Put
\[S(x)\,{:\!=}\,\sum_{n\leqslant x}\Delta(n),\quad {{\mathfrak S}}(x)\,{:\!=}\,\frac{1}{\log x}\sum_{n\leqslant x}{\Delta(n)\over n},\]
Footnote 1 so that (see [ Reference Hall and Tenenbaum12 , theorem. 61])
\[S(x)\ll x{{\mathfrak S}}(x)\qquad (x\geqslant 2).{}^1\]
After the works of Erdős [ Reference Erdős8 ], Hall—Tenenbaum [ Reference Hall and Tenenbaum11 , Reference Hall and Tenenbaum12 ], Hooley [ Reference Hooley15 ], Tenenbaum [ Reference Tenenbaum28 ], Koukoulopoulos-Tao [ Reference Koukoulopoulos and Tao17 ], Ford-Koukoulopoulos-Tao [ Reference Ford, Koukoulopoulos and Tao9 ], and La Bretèche-Tenenbaum [ Reference de la Bretèche and Tenenbaum4 ], we are now equipped with the boundsFootnote 2
\begin{equation}(\log_2x)^{1+\eta+o(1)}\ll {{\mathfrak S}}(x)\ll (\log_2x)^{5/2}\qquad (x\geqslant 3), \end{equation}
where
$\eta\approx0.353327$
is the exponent appearing in the new lower bound for the normal order of
$\Delta(n)$
by Ford, Green and Koukoulopoulos [
Reference Ford, Green and Koukoulopoulos7
].
The fact that the asymptotic order of magnitude of
${{\mathfrak S}}(x)$
is a power of
$\log_2x$
makes it pertinent to refine the estimates relating to the various applications, as the number of representations of an integer as a sum of powers, the Diophantine approximation of an irrational number by a rational with square denominator, or Waring’s problem.
Two other applications are also worth mentioning. First note that [
Reference Nair and Tenenbaum19
, corollary 8] readily yields that, for any given
$\varepsilon\gt 0$
, and uniformly for
$x\geqslant 3$
,
$1\leqslant |a|\leqslant x$
,
$x^\varepsilon\lt y\leqslant x$
, we have
\[\sum_{x\lt p\leqslant x+y}\Delta(|p+a|)\ll{|a|y{{\mathfrak S}}(x)\over \varphi(|a|) \log x}\ll{|a|y(\log_2x)^{5/2}\over \varphi(|a|)\log x},\]
where the letter p denotes a prime number. Second, observe that the upper bound of (1·1) furnishes, for any fixed
$\varepsilon\gt 0$
and uniformly for
$ 1\leqslant a \leqslant q\leqslant x^{2-\varepsilon}$
,
$(a,q)=1$
,
$x\geqslant 3$
, the estimate
\begin{equation}\sum_{\substack{n,m\leqslant x\\ mn {\equiv {a}\,({\mathrm{mod}\,}{q})} }} 1\ll {x^2\over q}(\log_2x)^{5/2}.\end{equation}
Indeed, if
$N\leqslant x^{1-\varepsilon/2}$
, a trivial bound, using
$\tau(n)=n^{o(1)}$
suffices, and if
$x^{1-\varepsilon/2}\lt N\leqslant x$
, we have
\begin{equation*}\sum_{\substack{ m\leqslant x\\ {N\lt n\leqslant 2N}\\ {mn{\equiv a\,({\rm mod\,} q)} }}} 1\leqslant \sum_{0\leqslant k\leqslant 2Nx/q} \Delta(kq+a)\ll {Nx\over q} {{\mathfrak S}}(x) ,\end{equation*}
where the latter bound results from a special case of [ Reference Nair and Tenenbaum19 , theorem 1]. This implies (1·2) by summing over N.
We intend here to return to the problem of sums of powers, also addressed in [ Reference Koukoulopoulos and Tao17 ].
Given
${\boldsymbol{c}}\,{:\!=}\,\{c_j \}_{0\leqslant {\;j}\leqslant t},\,{\boldsymbol{\ell}}\,{:\!=}\,\{\ell_j \}_{0\leqslant {\;j}\leqslant t}\in({{\mathbb N}}^*)^{t+1}$
, consider the number
\[r(n) = r(n;\,{\boldsymbol{c}},{\boldsymbol{\ell}}) \]
of representations of a natural integer n in the form
\[n=\sum_{0\leqslant {\;j}\leqslant t} c_jm_j^{\ell_j} \qquad ({\boldsymbol{m}} \in {{\mathbb N}}^{t+1}).\]
In all the sequel, we shall assume without loss of generality that
$\{\ell_j \}_{0\leqslant {\;j}\leqslant t}$
is non-decreasing.
Under the condition
$\sum_{0\leqslant {\;j}\leqslant t}1/\ell_j=1$
, we trivially have
\[\qquad V_0(x;\,{\boldsymbol{c}},{\boldsymbol{\ell}})\,{:\!=}\,\{ n\leqslant x \,:\, r(n)\geqslant 1\}\leqslant x\qquad \qquad (x\geqslant 1).\]
The exact order of magnitude of
$V_0(x)=V_0(x;\,{\boldsymbol{c}},{\boldsymbol{\ell}})$
is only known in the case
${\boldsymbol{\ell}}=(2,2)$
. A famous conjecture (see, for instance, [
Reference Balog and Brӱdern1
], or [
Reference Wang31
]) is that
$V_0(x)\asymp x$
for
${\boldsymbol{\ell}}=(3,3,3)$
,
${\boldsymbol{c}}=(1,1,1)$
. The case
$t=2$
,
${\boldsymbol{c}}=(1,1,1)$
,
${\boldsymbol{\ell}}=(2,4,4) $
was studied by Hooley [
Reference Hooley15
], Tenenbaum [
Reference Tenenbaum29
], Robert [
Reference Robert21
], and Koukoulopoulos–Tao [
Reference Koukoulopoulos and Tao17
]. Robert [
Reference Robert21
] also considered the case
${\boldsymbol{\ell}}=(2,3,6)$
.
We obtain the following result.
Theorem 1·1. Let
$t {\kern-1pt}\geqslant{\kern-1pt} 2$
,
${\boldsymbol{c}}{\kern-1pt}\in{\kern-1pt}({{\mathbb N}}^*)^{t+1}$
,
${\boldsymbol{\ell}}{\kern-1pt}={\kern-1pt}\{\ell_j \}_{0\leqslant {\;j}\leqslant t}$
,
$\ell_0{\kern-1pt}={\kern-1pt}2$
,
$\ell_1{\kern-1pt}\in{\kern-1pt} \{ 3,4\}$
,
$\sum_{1\leqslant {\;j}\leqslant t} 1/\ell_j{\kern-0pt}={\kern-1pt}{1}/{2}$
. Let
$s\,{:\!=}\,\max\{j\in[1,t-1]\,:\,\ell_{t-j+1}=\ldots=\ell_t\}$
. Assume:
-
(i)
$1\leqslant s\leqslant 3$
; -
(ii)
$\ell_t\geqslant 26$
if
$s=3$
; -
(iii)
$\sum_{j\geqslant r} 1/\ell_j\leqslant 1/\ell_{r-1}\ (1\leqslant r\leqslant t-s+1).$
Then, we have
\begin{equation}V_0(x;\,{\boldsymbol{c}},{\boldsymbol{\ell}})\gg x/{{\mathfrak S}}(x)\gg x/(\log_2x)^{5/2}.\end{equation}
We have not sought to specify uniformity in respect to the coefficients
$c_j$
in this statement.
Observe that the following sets fulfill the assumptions:
\begin{align*}&t=2,\,{\boldsymbol{\ell}}\in \{ (2,3,6) ,\, (2,4,4) \}; \\ &t=3,\,{\boldsymbol{\ell}}\in \{ (2,3,7,42),\, (2,3,8,24) , \, (2,3,9,18), \, (2,3,10,15),\, (2,3,12,12),\, (2,4,8,8) \};\\ &t=4,\,{\boldsymbol{\ell}}\in\{ (2,3,7,84,84), \, (2,3,8,48, 48), \, (2,4,8,16,16),\, (2,3,12,24,24)\}; \\ &t=5,\,{\boldsymbol{\ell}}\in\{(2,3,12,36,36,36),\, (2,3,10,45,45,45),\, (2,4,8,16,32,32)\}; \end{align*}
this list being non exhaustive. New sets can indeed be constructed by increasing the cardinality t. For instance, when
${\boldsymbol{\ell}}$
belongs to the above list, we can define
${\boldsymbol{\ell}}^+\in({{\mathbb N}}^*)^{t+2}$
by setting
$\ell^+_j\,{:\!=}\,\ell_j$
for
$0\leqslant {\;j}\leqslant t-1$
,
$\ell_t^+\,{:\!=}\,2\ell_t$
,
$\ell_{t+1}^+=2\ell_t$
. Yet another possibility is to define
${\boldsymbol{\ell}}^*\in({{\mathbb N}}^*)^{t+3}$
with
$\ell^*_t=\ell_{t+1}^*=\ell_{t+2}^*\,{:\!=}\,3\ell_t$
provided
$\ell_t\geqslant 9$
. Thus,
$\boldsymbol{\ell}=(2,3,6)$
induces (2,3,12,12) and
$ (2,3,18,18, 18). $
The article [
Reference Koukoulopoulos and Tao17
] comes back on the case
${\boldsymbol{\ell}}=(2,4,4)$
—see, however, the remark following the statement of Proposition 5·1 infra.
The cases with
$t\geqslant 3$
in Theorem 1·1 are new. The corresponding results rely on the bound
\begin{equation}\bigg| {\bigg\{(m_{t-2}, m_{t-1}, m_{t},n_{t-2}, n_{t-1}, n_{t})\,:\sum_{t-2\leqslant {\;j}\leqslant t} c_jm_j^{\ell_t} =\sum_{t-2\leqslant {\;j}\leqslant t} c_jn_j^{\ell_t}\leqslant x\bigg\}}\bigg|\ll x^{3/\ell_t}, \end{equation}
established by Salberger provided
$\ell_t\geqslant 26$
—see the proof of Proposition 5·2 infra for precise references. In private communication [
Reference Salberger24
], Salberger informed us that, as a direct consequence of his works [
Reference Salberger22
,
Reference Salberger23
], the bound (1·4) persists under the weaker condition
$\ell_t\geqslant 16$
. Assuming this, we get that (1·3) also holds for
${\boldsymbol{\ell}}=(2,3,18,18,18)$
and
${\boldsymbol{\ell}}=(2,4,8,24,24,24)$
. The starting point of the proof of Theorem 1·1 is classically the Cauchy-Schwarz inequality
\begin{equation}V_1(x)^2\leqslant V_0(x)V_2(x), \end{equation}
where we have set
\begin{equation} V_j(x)=V_j(x;\,{\boldsymbol{c}},{\boldsymbol{\ell}})\,{:\!=}\,\sum_{n\leqslant x} r(n)^j\qquad (\;j=1,2). \end{equation}
We shall deduce Theorem 1·1 from a general estimate regarding
$j=2$
.
2. Notation
Some notation must be introduced in order to state our results.
Given
$k\in{{\mathbb N}}^*$
,
${\boldsymbol{a}}=(a_j)_{1\leqslant {\;j}\leqslant k},{\boldsymbol{b}}=(b_j)_{1\leqslant {\;j}\leqslant k}\in{{\mathbb N}}^{*k}$
, we put
\[{{\mathfrak s}}{\boldsymbol{a}}\,{:\!=}\,\sum_{1\leqslant {\;j}\leqslant k}a_j,\quad \wp{\boldsymbol{a}}\,{:\!=}\,a_1\cdots a_k,\quad {\boldsymbol{a}}{\boldsymbol{b}}=(a_1b_1,\ldots,a_kb_k).\]
For each value of the parameters
$A\geqslant 1$
,
$B\geqslant 1$
,
$\varepsilon\gt 0$
, we designate by
${\mathcal{M}}_k(A,B,\varepsilon)$
the class of those non-negative arithmetic functions
$F\,:\,{{\mathbb N}}^{*k}\to{{{{{\mathbb R}}}}}^+$
, satisfying the condition
\begin{equation}F({\boldsymbol{a}}{\boldsymbol{b}})\leqslant \min\big\{A^{\Omega(\wp{\boldsymbol{a}})}, B(\wp{\boldsymbol{a}})^\varepsilon \big\}F({\boldsymbol{b}})\end{equation}
for all
${\boldsymbol{a}},{\boldsymbol{b}}\in{{\mathbb N}}^{*k}$
such that
$(\wp{\boldsymbol{a}}, \wp{\boldsymbol{b}})=1$
. By convention, the functions
$F \in {\mathcal{M}}_k(A, B, \varepsilon)$
are extended to
$\mathbb{N}^k$
by setting
$F({\boldsymbol{a}}) = 0$
if
$\min_j a_j =0$
. When
$F\neq0$
, we define
\begin{equation}G({\boldsymbol{a}})=G_F({\boldsymbol{a}})\,{:\!=}\,\max_{\substack{{\boldsymbol{b}}\in{{\mathbb N}}^{*k}\\{(\wp{\boldsymbol{a}},\wp{\boldsymbol{b}})=1}\\{F({\boldsymbol{b}})\neq0}}}{F({\boldsymbol{a}}{\boldsymbol{b}})\over F({\boldsymbol{b}})}\qquad \big({\boldsymbol{a}}\in{{\mathbb N}}^{*k}\big).\end{equation}
Given a family
$\{Q_j\}_{j=1}^{k}\in{{\mathbb Z}}[X_1, \ldots, X_t]^k$
of polynomials in t variables, we put
\begin{equation}Q=\prod_{1\leqslant {\;j}\leqslant k}Q_j=\prod_{1\leqslant h\leqslant r}R_h^{\gamma_h}\in{{\mathbb Z}}[X_1, \ldots, X_t], \quad g\,{:\!=}\,\deg Q,\end{equation}
where the
$R_h$
are irreducible in
${{\mathbb Z}}[X_1, \ldots, X_t]$
. We may then write, canonically,
\[Q_j=\prod_{1\leqslant h\leqslant r}R_h^{\gamma_{jh}}\qquad (1\leqslant {\;j}\leqslant k),\]
where
$\gamma_{jh}\geqslant 0$
for all j, h, so that
\[\gamma_h=\sum_{1\leqslant {\;j}\leqslant k}\gamma_{jh}\qquad (1\leqslant h\leqslant r), \qquad \sum_{1\leqslant h\leqslant r} \gamma_h \deg R_h =g.\]
Recall that a polynomial is said to be primitive if the greatest common divisor of its coefficients is 1. In all the sequel we assume the form Q to be primitive, which implies that the same holds for all the
$Q_j$
.
For
$T\in{{\mathbb Z}}[X_1, \ldots, X_t]$
, we write
\begin{equation}\varrho_T^+(s)\,{:\!=}\,\sum_{\substack{{\boldsymbol{\xi}}\in[1,s]^t\\{T({\boldsymbol{\xi}}){\equiv 0\,({\rm mod\,}s)} }}}1\qquad (s\geqslant 1).\end{equation}
Letting
$\kappa(s)\,{:\!=}\,\prod_{p|s}p$
denote the squarefree kernel of a natural integer s, we further put
\begin{equation}{\mathcal{K}}({\boldsymbol{s}})\,{:\!=}\,{\mathrm{lcm}}(s_j\kappa(s_j))_{1\leqslant {\;j}\leqslant r}=[s_1\kappa(s_1),\ldots,s_r\kappa(s_r)]\qquad \big({\boldsymbol{s}}=(s_1,\ldots,s_r)\in{{\mathbb N}}^{*r}\big).\end{equation}
Given
${{{\boldsymbol{R}}}}\in{{\mathbb Z}}[X_1,\ldots,X_t]^r$
, we may then define two arithmetic functions of the variable
${\boldsymbol{s}}=(s_1,\ldots,s_r)\in{{\mathbb N}}^{*r}$
by
\begin{align} \varrho^+_{{{\boldsymbol{R}}}}({\boldsymbol{s}})\,{:\!=}\,\sum_{\substack{{\boldsymbol{\xi}}\in [1,\wp{\boldsymbol{s}}]^t\\R_h({\boldsymbol{\xi}}){\equiv 0\,({\rm mod\,}{s_h})}\, (1\leqslant h\leqslant r)}}1,\end{align}
\begin{align} \varrho^\#_{{{\boldsymbol{R}}}}({\boldsymbol{s}})\,{:\!=}\,\sum_{\substack{{\boldsymbol{\xi}}\in[1,{\mathcal{K}}({\boldsymbol{s}})]^t \\ s_h\|R_h({\boldsymbol{\xi}})\, (1\leqslant h\leqslant r)\\ (R_h({\boldsymbol{\xi}})/s_h,\wp {\boldsymbol{s}})=1 }}1, \end{align}
where the symbol
$a\| b$
means that the conditions
$a|b$
and
$(a,b/a)=1$
hold simultaneously. It is worthwhile to note that the above definition of
$\varrho^\#_{{{\boldsymbol{R}}}}$
is more restrictive than that of [
Reference de la Bretèche and Tenenbaum3
]. The grounds for this alteration are described in [
Reference Henriot14
].
For any
${\boldsymbol{s}}=(s_1,\ldots,s_r)\in{{\mathbb N}}^{*r}$
, let us put
\[s'_j\,{:\!=}\,\prod_{1\leqslant h\leqslant r}s_h^{\gamma_{jh}}\quad (1\leqslant {\;j}\leqslant k),\quad {\boldsymbol{s}}'\,{:\!=}\,(s'_1,\ldots,s_k'),\qquad s''\,{:\!=}\,\wp{\boldsymbol{s}}'=\prod_{1\leqslant {\;j}\leqslant k}s'_j=\prod_{1\leqslant h\leqslant r}s_h^{\gamma_h}.\]
Thus, given
${\boldsymbol{m}}=(m_1,\ldots, m_t)\in{{\mathbb N}}^{*t}$
, and writing
$s_h=R_h({\boldsymbol{m}})\ (1\leqslant h\leqslant r)$
, we have
\[s'_j=Q_j({\boldsymbol{m}})\quad (1\leqslant {\;j}\leqslant k),\quad s''=Q({\boldsymbol{m}}).\]
With these pieces of notation, to any function F in
${\mathcal{M}}_k(A,B,\varepsilon)$
we may associate a further function
$ \widehat{F}$
in
${\mathcal{M}}_r(A^g, B,g\varepsilon)$
defined by
\[\widehat{F}({\boldsymbol{s}})=F({\boldsymbol{s}}').\]
We then designate by
$\widehat{G}\,{:\!=}\,G_{\widehat{F}}$
the function associated to
$\widehat{F}$
by (2·2) with
$k=r$
and, for
$1\leqslant h\leqslant r$
, we let
$\widehat{G}_h\,:\,{{\mathbb N}}^*\to{{{{\mathbb{R}}}}}^+$
denote the composition of
$\widehat{G}$
with the hth coordinate.
3. Statement of results
The key tool of this work consists in an estimate of the quantity
\begin{equation*}\begin{aligned}S\,{:\!=}\,&\sum_{x_j\lt n_j\leqslant x_j+y_j \, (1\leqslant {\;j}\leqslant t)}F\big(|Q_1({{{\boldsymbol{n}}}})|,\ldots, |Q_k({{{\boldsymbol{n}}}})|\big)\\=&\sum_{x_j\lt n_j\leqslant x_j+y_j \, (1\leqslant {\;j}\leqslant t)}\widehat{F}\big(R_1({{{\boldsymbol{n}}}}),\ldots, R_r({{{\boldsymbol{n}}}})\big).\end{aligned}\end{equation*}
The following result is obtained. It is the t-dimensional counterpart of that of [
Reference Henriot13
]. We let
$\|Q\|$
denote the maximum of the absolute values of the coefficients of a real form Q.
Theorem 3·1. Let
$k,t\in{{\mathbb N}^*}$
, and let
$\{Q_j\}_{j=1}^{k}\in {{\mathbb Z}}[X_1, \ldots, X_t]^k$
be a family of primitive polynomials. Define
$Q\in{{\mathbb Z}}[X_1, \ldots, X_t]$
,
$\{R_h\}_{1\leqslant h\leqslant r}\in{{\mathbb Z}}[X_1, \ldots, X_t]^r$
,
$g\in{{\mathbb N}}$
by (2·3),
$\varrho_Q^+$
by (2·4), and
$\varrho_{{{\boldsymbol{R}}}}^\#$
by (2·7). For all
\begin{equation}\alpha\in]0,1],\quad \beta\in ]0,1[,\quad A\geqslant 1,\quad B\geqslant 1, \quad 0\leqslant \varepsilon\leqslant \alpha \beta/ \{50g^2( \beta g+1)\},\end{equation}
and uniformly under the conditions
\begin{equation}\begin{aligned}F&\in {\mathcal{M}}_k(A,B,\varepsilon),\quad {\boldsymbol{x}}=(x_1,\ldots,x_t)\in{{\mathbb N}}^{*t},\quad {\boldsymbol{y}}=(y_1,\ldots,y_t)\in{{\mathbb N}}^{*t},\\x&\, {:\!=}\min_{1\leqslant {\;j}\leqslant t} x_j\geqslant c\big\{\max_{1\leqslant {\;j}\leqslant t}x_j+{\Vert{Q}\Vert}\big\}^{ \beta},\quad x_j^\alpha\leqslant y_j\leqslant x_j \quad (1\leqslant {\;j}\leqslant t),\end{aligned}\end{equation}
we have
\begin{equation}\sum_{\substack{{{{\boldsymbol{n}}}}\in{{\mathbb N}}^{*t}\\{x_j-y_j\lt n_j\leqslant x_j (1\leqslant {\;j}\leqslant t)}}}F\big(|Q_1({{{\boldsymbol{n}}}})|,\ldots, |Q_k({{{\boldsymbol{n}}}})|\big) \ll \wp{\boldsymbol{y}} E_{{{\boldsymbol{R}}}}({{\mathfrak s}} {\boldsymbol{x}})\prod_{g\lt p\leqslant x}\Big(1-{\varrho_Q^+(p)\over p^t}\Big),\end{equation}
where c and the implicit constant depend at most upon g,
$\alpha$
,
$ \beta$
, A, B and where we have set
\begin{equation} E_{{{\boldsymbol{R}}}}(v)\,{:\!=}\,\sum_{\substack{{\boldsymbol{s}}\in {{\mathbb N}}^{*r}\\{\wp{\boldsymbol{s}}\leqslant v}}}\widehat{F}({\boldsymbol{s}}){\varrho_{{{\boldsymbol{R}}}}^\#({\boldsymbol{s}})\over {\mathcal{K}}({\boldsymbol{s}})^t}\qquad (v\geqslant 1).\end{equation}
Remarks.
-
(i) We have
(3·5)however the sum
\begin{equation}{ \varrho^\#_{{{\boldsymbol{R}}}}({\boldsymbol{s}})\over {\mathcal{K}}({\boldsymbol{s}})^t}\leqslant {{\varrho^+_{{{\boldsymbol{R}}}}}({\boldsymbol{s}})\over (\wp{\boldsymbol{s}})^t}\quad \qquad \big({\boldsymbol{s}}=(s_1,\ldots,s_r)\in{{{\mathbb N}}^{*r}}\big),\end{equation}
$ E_{{{\boldsymbol{R}}}}(x)$
may turn out to be smaller than that which follows from this upper bound. For instance, if
$k=1$
,
$t=1$
,
$Q(X)=Q_1(X)=X(X+\ell)$
, we have
$\varrho^\#_{{{\boldsymbol{R}}}}(p)=0$
as soon as
$p\mid \ell$
, while
$\varrho^+_{{{\boldsymbol{R}}}}(p)=1$
. As noted in [
Reference Henriot13
] and [
Reference Henriot14
], this can be significant when it comes to obtaining results uniform with respect to
$\ell$
.
-
(ii) A weaker version of (3·3) for
$k=1$
has recently been established [
Reference Chan, Koymans, Pagano and Sofos6
, theorem 1·15], where, in definition (3·4), the left-hand side of (3·5) is replaced by its majorant. Note that the dependence of our bounds with respect to the coefficients of Q is effective. -
(iii) In the case
$t=1$
, Theorem 3·1 is due to Henriot [
Reference Henriot13
], [
Reference Henriot14
], extending [
Reference Shiu26
], then [
Reference Nair18
] and [
Reference Nair and Tenenbaum19
]. When the
$Q_j$
are binary forms in
${{\mathbb Z}}[X_1,X_2]$
, Theorem 3·1 has been proved in [
Reference de la Bretèche and Tenenbaum3
] with a slightly different definition of
$\varrho^\#$
, refining and extending the estimates of [
Reference de la Bretèche and Browning2
]. -
(iv) When F is multiplicative and satisfies an assumption such that
$F({{{\boldsymbol{n}}}})\geqslant \eta^{\Omega(\wp{{{\boldsymbol{n}}}})}$
for some
$\eta\gt 0$
, it is possible to establish lower bounds for the the left-hand side of (3·3) of the same order of magnitude as the right-hand side. We refer to [
Reference Henriot13
] for this aspect of the question.
4. Proof of Theorem 3·1
Recall a consequence of the Schwartz-Zippel lemma—see [ Reference Schwartz27 , lemma 1], or [ Reference Zippel32 , theorem 1].
Proposition 4·1. Let
$t\geqslant 1$
and let
$Q\in {{\mathbb Z}}[X_1, \ldots, X_t] $
be a primitive polynomial of degree g. For all prime numbers p, we have
\begin{equation}\varrho_Q^+(p )\leqslant g p^{ t-1 }.\end{equation}
In particular, for all integers
$\nu\geqslant 1$
, we have
$\varrho_Q^+(p^\nu)\leqslant g p^{\nu t-1}\lt p^{\nu t}$
provided
$p\gt g$
.
Let c(Q) denote the content of the polynomial Q of t variables and let
$v_p(n)$
denote the p-adic valuation of an integer n. The following lemma is an effective version of [
Reference Chan, Koymans, Pagano and Sofos6
, lemma 2·8]—see also [
Reference Pierce, Schindler and Wood20
, lemma 4·10].
Lemma 4·2. Let
$t\geqslant 1$
and let
$Q\in {{\mathbb Z}}[X_1, \ldots, X_t] $
be a polynomial of degree g. For all prime numbers p and all integers
$\nu\geqslant 1$
, we have
\begin{equation}\varrho_Q^+(p^\nu)\leqslant g^t(\nu+1)^{t-1} p^{\nu (t-1/g)+ v_p(c(Q))/g}.\end{equation}
Proof. Consider first the case
$t=1$
. When
$(c(Q),p)=1$
, inequality (4·2) has been proved by Stewart [
Reference Stewart25
]. If
$\gamma=v_p(c(Q)) \geqslant 1 $
, we have
\[\varrho_Q^+(p^\nu)=p^\gamma \varrho_{Q^*}^+(p^{\nu-\gamma}) \leqslant g p^{\nu (1-1/g)}p^{v_p(c(Q))/g},\]
with
$Q^*\,{:\!=}\,Q/p^\gamma$
. Thus (4·2) holds for
$t=1$
.
We now proceed by induction on t. Let
$x_1$
be such that Q depends on
$x_1$
and write
\[Q(x_1,x_2,\ldots, x_t)=\sum_{0\leqslant {\;j}\leqslant g} Q_j( x_2,\ldots, x_t)x_1^{\;j} \]
with
$Q_j \in {{\mathbb Z}}[X_2, \ldots, X_t] $
and
$g_j\,{:\!=}\,\deg Q_j\leqslant g-j$
. By the case
$t=1$
, we have
\[\varrho_Q^+(p^\nu)\leqslant g p^{\nu (1-1/g)}\sum_{0\leqslant \gamma\leqslant \nu} p^{\gamma/g+(t-1)(\nu-\gamma)} N(\gamma,p),\]
with
$N(\gamma,p)\,{:\!=}\,|\{ {\boldsymbol{x}}\in ( {{\mathbb Z}}/p^\gamma{{\mathbb Z}})^{t-1}\,:\, Q_j({\boldsymbol{x}}){\equiv 0\,({\rm mod\,}{p^\gamma})}\ (0\leqslant {\;j}\leqslant g)\}|$
.
Now the induction hypothesis yields
\begin{align*} N(\gamma,p)& \leqslant \min_{0\leqslant {\;j}\leqslant g}|\{ {\boldsymbol{x}}\in ( {{\mathbb Z}}/p^\gamma{{\mathbb Z}})^{t-1} \,:\, Q_j({\boldsymbol{x}}){\equiv 0\,({\rm mod\,}{p^\gamma})} \}| \cr& \leqslant \min_{0\leqslant {\;j}\leqslant g} g_j^{t-1}(\nu+1)^{t-2} p^{\gamma (t-1)+\min\{ v_p(c(Q_j))-\gamma, 0\}/g_j} \cr& \leqslant g^{t-1}(\nu+1)^{t-2} p^{\gamma (t-1-1/g )+v_p(c(Q))/g }, \end{align*}
since
$ \min_{0\leqslant {\;j}\leqslant g} v_p(c(Q_j))=v_p(c(Q))$
. This completes the proof of (4·2).
Let us now state the analogue in our context of [
Reference Henriot13
, lemma 6]. This is a straightforward consequence of the fundamental lemma of the combinatorial sieve. It coincides with [
Reference de la Bretèche and Tenenbaum3
, lemma 3·3] when
$t=2$
and with [
Reference Henriot13
, lemma 6] when
$t=1$
. As in [
Reference Henriot13
, lemma 6], we let
$\alpha$
be an arbitrary constant in ]0,1] and define
\begin{equation} {{\alpha_1\,{:\!=}\,\tfrac3{25} \alpha,\quad \alpha_2\,{:\!=}\,\varepsilon_1/6g}}. \end{equation}
Lemma 4·3. Let
$t\geqslant 1$
,
$0\lt \alpha\lt 1$
, and let
$Q, R_h\in{{\mathbb Z}}[X_1, \ldots, X_t]$
be polynomials satisfying the hypotheses of Theorem 3·1. Under the conditions
\[{\boldsymbol{a}}\in{{\mathbb N}}^{*r},\ {\boldsymbol{x}},{\boldsymbol{y}}\in{{\mathbb N}}^{*t},\ w\,{:\!=}\,\min_jx_j,\ \wp{\boldsymbol{a}}\leqslant w^{\alpha_1},\ z\leqslant w^{\alpha_2},\ x_j^{\alpha}\leqslant y_j\leqslant x_j,\]
we have
\begin{equation}\sum_{\begin{smallmatrix}{x_j\lt n_j\leqslant x_j+y_j \, (1\leqslant {\;j}\leqslant t)}\\{a_h\| R_h({{{\boldsymbol{n}}}})\ (1\leqslant h\leqslant r)}\\{(p\mid Q({{{\boldsymbol{n}}}}), \, g\lt p\leqslant z)\Rightarrow p\mid \wp{\boldsymbol{a}}}\\{(R_h({{{\boldsymbol{n}}}})/a_h,\wp{\boldsymbol{a}})=1\,(1\leqslant h\leqslant r)}\end{smallmatrix}}1\ll\wp{\boldsymbol{y}}{\varrho_{{{\boldsymbol{R}}}}^\#({\boldsymbol{a}})\over {\mathcal{K}}({\boldsymbol{a}})^t}\prod_{\substack{g\lt p\leqslant z\\{p\,\nmid\, \wp{\boldsymbol{a}}}}}\Big(1-\frac{\varrho_Q^+(p)}{p^r}\Big).\end{equation}
Note that, since the product in (3·3) only covers primes
$p\gt g$
, it avoids any fixed prime divisor of the
$R_h$
. It is also worthwhile to keep in mind that
\begin{equation}{\varrho_{{{\boldsymbol{R}}}}^\#({\boldsymbol{a}})\over {\mathcal{K}}({\boldsymbol{a}})^t}={\mathrm{dens}} \Big\{ {{{\boldsymbol{n}}}}\in {{\mathbb Z}}^t\,:\, a_h\parallel R_h({{{\boldsymbol{n}}}})\ (1\leqslant h\leqslant r), \, (p\mid \wp{\boldsymbol{a}}, \, p\nmid a_h)\Rightarrow (p\nmid R_h({{{\boldsymbol{n}}}})) \Big\} .\end{equation}
Proof. In the above statement, the parameters
$\alpha_1$
and
$\alpha_2$
defined in (4·3) coincide respectively with
$\varepsilon_1$
and
$\varepsilon_3$
from [
Reference Henriot13
, lemma 6]. Lemma 4·3 follows by applying the combinatorial sieve to the sequence
\[\mathcal{A} \,{:\!=}\,\left\{ Q(n_1,\ldots, n_t)\,: \begin{aligned} &x_j\lt n_j\leqslant x_j+y_j \ (1\leqslant {\;j}\leqslant t),\ a_h\| R_h({{{\boldsymbol{n}}}})\ (1\leqslant h\leqslant r),\\&(p\mid \wp{\boldsymbol{a}}/ a_h, \, p\,\nmid\,a_h)\Rightarrow (p\,\nmid\, R_h(n))\end{aligned}\right\}.\]
Thus, we can select
$A_1=g+1$
in the hypotheses of the sieve—see [
Reference Halberstam and Richert10
, lemma 2·2], condition
$(\Omega_1)$
— since the upper bound (4·1) implies that
\[{\varrho_Q^+(p)\over p^t}\leqslant {g \over p}\leqslant 1-{1\over g+1}\quad \big(p\geqslant g+1,\,p\nmid \wp{\boldsymbol{a}}\big).\]
Further details are omitted.
Substituting Lemma 4·2 to [
Reference Henriot13
, lemma 6] in the proof of [
Reference Henriot13
, theorem 5] readily furnishes estimate (3·3). Note that Lemma 4·2 is crucial in order to get effective bounds in terms of
${\Vert{Q}\Vert}$
.
5. Proof of Theorem 1·1
5·1. Statements
It follows from (1·6) that
\[V_2(x;\,{\boldsymbol{c}},\,{\boldsymbol{\ell}})={\bigg|\bigg\{ ({\boldsymbol{m}},{{{\boldsymbol{n}}}})\in ({{\mathbb N}}^*)^{t+1 }\times ({{\mathbb N}}^*)^{t+1 }: \sum_{0\leqslant {\;j}\leqslant t}c_jm_j^{\ell_j}= \sum_{0\leqslant {\;j}\leqslant t}c_j n_j^{\ell_j}\leqslant x\bigg\}\bigg|},\]
and so
$V_2(x)=V_2(x;\,{\boldsymbol{c}},\,{\boldsymbol{\ell}})=V_2^{\neq}(x)+V_2^{=}(x)$
, where the subsums
$V_2^{\neq}(x)$
and
$V_2^{=}(x)$
respectively correspond to the extra conditions
$m_0\neq n_0$
and
$m_0=n_0$
. We obtain the following estimates.
Proposition 5·1. Let
$t\geqslant 1$
,
$\delta\gt 0$
,
${\boldsymbol{c}},\,{\boldsymbol{\ell}}\in({{\mathbb N}}^*)^{t+1}$
, satisfy
$\ell_0\geqslant 2$
and
$\delta=\sum_{1\leqslant {\;j}\leqslant t} 1/\ell_j $
. Then
\begin{equation} V^{\neq}_2(x)\ll x^{2\delta}{{\mathfrak S}}(x)\ll x^{2\delta}(\log_2x)^{5/2} \qquad (x\geqslant 3). \end{equation}
The second bound follows from (1·1). The exponent
$5/2$
hence measures, in the current state of knowledge, the deviation of this general estimate from optimality. In [
Reference Koukoulopoulos and Tao17
, Section 8], Koukoulopoulos and Tao indicate the exponent
$=2^{4L}+15/4$
for
$L=\max\ell_j$
,
$\ell_0=2$
,
$\delta={1}/{2}$
. The condition
$m_0\neq n_0$
has been omitted in [
Reference Koukoulopoulos and Tao17
]. Since it is necessary for a general result, the argument given in [
Reference Koukoulopoulos and Tao17
] only yields the bound [
Reference Koukoulopoulos and Tao17
, (1·5)] in the historical case
${\boldsymbol{\ell}}=(2,4,4)$
considered by Hooley. The exponent of
$\log_2x$
stated in [
Reference Koukoulopoulos and Tao17
] is then
$65539.75.$
Our approach to Theorem 1·1 is based on an inductive argument described in the following statement.
Proposition 5·2. Let
$t\geqslant 2$
,
${\boldsymbol{c}} {=\{c_j\}_{j=0}^{t}\in({{\mathbb N}}^*)^{t+1},{\boldsymbol{\ell}}=\{\ell_j\}_{j=0}^{t}\in({{\mathbb N}}^*)^{t+1}}$
with
$\ell_0\geqslant 2$
. Put
$\check{{\boldsymbol{c}}}\,{:\!=}\,(c_1,\ldots, c_t)$
,
$\check{{\boldsymbol{\ell}}}=(\ell_1,\ldots, \ell_t)$
. We have
\begin{equation}V^{=}_2(x;\,{\boldsymbol{c}},\,{\boldsymbol{\ell}})\leqslant \Big({x\over c_0}\Big)^{1/\ell_0} V_2(x;\,\check{{\boldsymbol{c}}},\check{{\boldsymbol{\ell}}}) \qquad (x\geqslant 3).\end{equation}
Moreover, if
$t\in\{1,2\}$
and
$\ell_{t-1}=\ell_t\geqslant 3$
, or if
$ t=3$
and
$\ell_{t-2}=\ell_{t-1}=\ell_t\geqslant 26$
, then
\begin{equation}V_2(x;\,\check{{\boldsymbol{c}}},\check{{\boldsymbol{\ell}}})\ll x^{t/\ell_t }.\end{equation}
Proof. The bound (5·2) is clear, and so is (5·3) for
$t=1$
. The remaining bounds (5·3) are due to Salberger: for
$t=2$
, see [
Reference Salberger23
, cor. 0.7] ; for
$t=3$
,
$\ell_t\geqslant 33$
, see [
Reference Browning and Heath-Brown5
] ; for
$t=3$
,
$\ell_t\geqslant 26 $
, see [
Reference Salberger22
, cor. 4·6].
As mentioned in the introduction, we have been informed by Salberger that he intends to write up the proof that (5·2) holds for
$t=3$
,
$\ell_t\geqslant 16$
, and hopefully
$\ell_t\geqslant 12$
.
5·2. A lemma from the literature
Define
$P^+(n)$
as the largest prime factor of an integer n, with the convention that
$P^+(1)\,{:\!=}\,1$
. The following statement is established in [
Reference Henriot13
, lemmas 2 and 5].
Lemma 5·3. Let
$r, g\geqslant 1$
,
$A\geqslant 1$
,
$B\geqslant 1$
,
$\varepsilon\gt 0$
and let
$\{\sigma_j\}_{j=1}^r$
,
$\{\vartheta_j\}_{j=1}^r$
be two families of multiplicative arithmetical functions. Assume that, uniformly for prime powers
$p^\nu$
, we have
$\sigma_j(p^\nu)\ll 1$
,
$\vartheta_j(p^\nu )=1+O(1/p)$
. Let
$\widehat{F}\in{\mathcal{M}}_r(A^g, B,g\varepsilon)$
, and define
\[H({\boldsymbol{s}}) \,{:\!=}\,\widehat{F}({\boldsymbol{s}}) \prod_{1\leqslant {\;j}\leqslant r} \sigma_j(s_j){\varrho_{{{\boldsymbol{R}}}}^\#({\boldsymbol{s}})\over {\mathcal{K}}({\boldsymbol{s}})^t}\qquad ({\boldsymbol{s}}\in {{\mathbb N}}^{*r}).\]
Then, for
$x\geqslant 3$
, we have
\begin{align} \sum_{\substack{ {\boldsymbol{s}}\in {{\mathbb N}}^{*r}\\ \wp{\boldsymbol{s}}\leqslant x}} H({\boldsymbol{s}})\prod_{1\leqslant {\;j}\leqslant r} \vartheta_j(s_j)\ll\sum_{\substack{ {\boldsymbol{s}}\in {{\mathbb N}}^{*r}\\ \wp{\boldsymbol{s}}\leqslant x}}H({\boldsymbol{s}}) ,\end{align}
\begin{align}\sum_{\substack{{\boldsymbol{s}}\in {{\mathbb N}}^{*r}\\P^+(\wp{\boldsymbol{s}})\leqslant x}} H({\boldsymbol{s}}) \ll \sum_{\substack{{\boldsymbol{s}}\in{{\mathbb N}}^{*r}\\ \wp{\boldsymbol{s}}\leqslant x}} H({\boldsymbol{s}}).\end{align}
5·3. Proof of Proposition 5·1.
Consider
\begin{equation}Q(X_1,\ldots, X_t, Y_1,\ldots, Y_t)\,{:\!=}\,\sum_{1\leqslant {\;j}\leqslant t} c_jX_j^{\ell_j}-\sum_{1\leqslant {\;j}\leqslant t} c_jY_j^{\ell_j} .\end{equation}
The following statement gathers the necessary estimates regarding function
$\varrho_Q$
defined in (2·4).
Lemma 5·4. Let
$t\geqslant 1$
,
${\boldsymbol{c}},\,{\boldsymbol{\ell}}\in{{\mathbb N}}^{* t}$
be such that
\[L\,{:\!=}\,\max_{1\leqslant {\;j}\leqslant t}\ell_j,\quad \gcd(c_1,\ldots,c_t)=1.\]
For all prime numbers p and all integers
$\nu\geqslant 1$
and Q defined by (5·6), we have
\[\varrho_Q(p )=p^{2t-1}+O(p^t), \qquad \varrho_Q(p^\nu)\leqslant L p^{ 2t\nu-1 } , \qquad \varrho_Q(p^\nu)\ll \nu p^{2(t-\delta)\nu}\]
with
$\delta\,{:\!=}\,\sum_{j=1}^t 1/\ell_j$
.
Proof. The second estimate follows from Lemma 4·1. The first formula is included in the first estimate of [
Reference Robert21
, lemma 3·4], itself resting on an estimate of Korobov [
Reference Korobov16
, chapter 1, theorem 5]. The third bound is a consequence of the second estimate in [
Reference Robert21
, lemma 3·4], resting on Korobov’s [
Reference Korobov16
, chapter. 1, theorem 6], observing that the proof given in [
Reference Robert21
] remains valid as it stands when
$\delta\lt 1/2$
.
We are now ready to complete the proof of Proposition 5·1. Put
$L\,{:\!=}\,\max \ell_j$
. Proposition 5·1 will follow from Theorem 3·1, applied with
$k=1$
,
$Q_1=Q$
,
$F=\Delta$
. Up to dividing through by
$\gcd(c_1,\ldots,c_t)$
, the
$c_j$
may be assumed coprime. The polynomial Q is then irreducible in
${{\mathbb Z}}[X_1,\ldots, X_t, Y_1,\ldots, Y_t]$
, so that
$r=1$
. Indeed, we may write
$Q=c_1X_1^{\ell_1}+Q_1$
, where
$Q_1\in {{\mathbb Z}}[X_2,\ldots X_t, Y_1,\ldots, Y_t]$
has no square factor in
${{\mathbb Q}}[X_2,\ldots, X_t, Y_1,\ldots, Y_t]$
. This enables applying Eisenstein’s criterion associated to a non constant irreducible factor of
$Q_1$
in
${{\mathbb Z}}[X_2,\ldots, X_t, Y_1,\ldots, Y_t]$
.
Taking the above into account, Lemma 5·4 furnishes
\[ \prod_{L \lt p\leqslant x}\Big(1-{\varrho_Q^+(p)\over p^{2t}}\Big)\asymp \log x\qquad (x\geqslant 2L). \]
Moreover, the quantity
$ E_Q(v)$
defined in (3·4) satisfies
\begin{equation} E_Q(v) \asymp \sum_{n\leqslant v} {\Delta(n)\over n}\qquad (v\geqslant 1). \end{equation}
This follows from Lemmas 5·3 and 5·4. Indeed, since
$k=r=1$
, we have
\begin{equation} E_Q(v)=\sum_{n\leqslant v} {\Delta(n)\varrho_Q^{\#}(n)\over {\mathcal{K}}(n)^{2t}}\cdot \end{equation}
By Lemma 5·4, there exist two multiplicative functions
$\vartheta_1$
and
$\vartheta_2$
such that:
\begin{equation*}\begin{aligned}\rm{(i)}&\quad \vartheta_j(p^\nu)=1+O(1/p)\quad (\;j\in \{ 1,2\},\ \nu\geqslant 1),\\\rm{(ii)}&\quad \varrho^{\#}(n)/{\mathcal{K}}(n)^{2t}=\varrho^{+}(n){\vartheta_1}(n)/n^{2t}=\vartheta_2(n)/n\quad (n\geqslant 1).\end{aligned}\end{equation*}
A first application of (5·4) provides the upper bound in (5·7). To establish the corresponding lower bound, we note that, by condition (i), we have
$\vartheta_2(p)\neq0$
for all but only a bounded number of primes p. Let
$\vartheta_3(p^\nu)=\vartheta_2(p^\nu)+{\textbf{1}}_{\{\vartheta_2(p^\nu)=0\}}(p^\nu)$
. Applying (5·5) to replace condition
$n\leqslant v$
by
$p|n\Rightarrow p\leqslant v$
, we hence see that substituting
$\vartheta_3$
to
$\vartheta_2$
in (5·8) does not alter the order of magnitude of the sum. A second application of (5·4) with
$H(n)\,{:\!=}\,\Delta(n)\vartheta_3(n)$
and
$\vartheta=1/\vartheta_3$
then furnishes the lower bound in (5·7).
We may now apply Theorem 3·1 in order to derive 5·1. Put
\[({\boldsymbol{m}},{{{\boldsymbol{n}}}})=(m_1,\ldots, m_t, n_1,\ldots, n_t).\]
By symmetry, we may assume
$n_0\gt m_0$
. Let us then split the variation range of
$m_0+n_0$
into dyadic intervals ]N,2N] with
$N=2^\nu\leqslant 2x^{1/\ell_0}$
. Up to neglecting a contribution
$\ll x^{2\delta}$
, we may assume
$N\gt x^{1/\ell_0-1/2\ell_0\ell_k}$
. Indeed, if
$N\leqslant x^{1/\ell_0-1/2\ell_0\ell_k}$
, then for all fixed
$(m_1,\ldots,m_k)$
,
$(n_1,\ldots,n_{k-1})$
, the power
$n_k^{\ell_k}$
lies in a fixed interval of length
$\ll N^{\ell_0}\ll x^{1-1/2\ell_k}.$
The number of admissible
$n_k$
is hence
$\ll x^{1/\ell_k-1/2\ell_k^2}.$
However, for each fixed
$n_k$
, the number of admissible pairs
$(m_0,n_0)$
is
$\ll x^{\varepsilon}$
for any
$\varepsilon\gt 0$
. This plainly implies that the contribution of those
$N\leqslant x^{1/\ell_0-1/2\ell_0\ell_k}$
is
$ \ll x^{2\delta-1/2\ell_k^2+\varepsilon}\ll x^{2\delta}$
provided that
$\varepsilon\leqslant 1/2\ell_k^2.$
If
$(m_0,{\boldsymbol{m}},n_0,{{{\boldsymbol{n}}}})$
is counted in
$V_2^{\neq}(x)$
, we have
$Q({\boldsymbol{m}},{{{\boldsymbol{n}}}})=c_0(n_0^{\ell_0}-m_0^{\ell_0})$
, hence the divisor
$n_0^{\ell_0-1}+m_0n_0^{\ell_0-1}+\ldots +m_0^{\ell_0-1}$
of
$Q({\boldsymbol{m}},{{{\boldsymbol{n}}}})$
is of size
$ N^{\ell_0-1}$
. As a consequence, for each 2t-tuple
$({\boldsymbol{m}},{{{\boldsymbol{n}}}})$
, the number of admissible pairs
$(m_0,n_0)$
does not exceed
$\Delta( Q({\boldsymbol{m}},{{{\boldsymbol{n}}}})).$
Now let us split the variation ranges of the
$m_j^{\ell_j}$
and
$n_h^{\ell_h}$
into intervals of size
$N^{\ell_0}$
, say
$ m_j^{\ell_j}\in ]r_jN^{\ell_0}, (r_j+1)N^{\ell_0}] $
and
$ n_h^{\ell_h}\in ]s_hN^{\ell_0}, (s_h+1)N^{\ell_0}] $
with
$r_j,s_h\ll x/N^{\ell_0}$
. For each
$r_j$
, the integer
$m_j$
is hence restricted to an interval of length
$\ll N^{{\ell_0}/\ell_j}/(r_j+1)^{(1-1/\ell_j)}.$
For each pair
$(r_j,s_j)$
, Theorem 3·1 furnishes a contribution
\begin{equation}\ll\prod_{ 1\leqslant {\;j}\leqslant t} {N^{2{\ell_0}/\ell_j } \over (r_j+1)^{1-1/\ell_j}(1+s_j)^{1-1/\ell_j}} {E_Q(x)\over \log x}\cdot\end{equation}
Here, the hypothesis
$N\gt x^{1/\ell_0-1/2\ell_0\ell_k}$
has been used in a crucial way since it implies that
$m_j$
lies in a range of size
$\gg x^{1/\ell_j-1/2\ell_k}\gg x^{ 1/2\ell_k} $
. This enables us to check hypotheses (3·2) of Theorem 3·1, which require that conditions (3·2) be satisfied by
\begin{equation*}\begin{aligned} &x_j=(r_j N^{\ell_0})^{1/\ell_j},\quad x_{j+k}=(s_j N^{\ell_0})^{1/\ell_j},\quad y_j=\{(r_j +1)N^{\ell_0}\}^{1/\ell_j}-(r_j N^{\ell_0})^{1/\ell_j},\\&y_{j+k}= ((s_j +1)N^{\ell_0})^{1/\ell_j}-(s_j N^{\ell_0})^{1/\ell_j},\end{aligned}\end{equation*}
for
$1\leqslant {\;j}\leqslant k.$
Thus, the choice
$\alpha=1/2\ell_k$
,
$\beta=1/\ell_k-1/2\ell_k^2$
, is admissible in (3·2).
Now observe that relations
$\big|m_j^{\ell_j}- n_j^{\ell_j}-(r_j-s_j)N^{\ell_0}\big|\leqslant N^{\ell_0}$
and
\[c_0|n_0^{\ell_0}-m_0^{\ell_0}|=\Big| \sum_{1\leqslant {\;j}\leqslant t}c_j (m_j^{\ell_j}- n_j^{\ell_j})\Big|\leqslant 2^{\ell_0}c_0N^{\ell_0}\]
imply
$\big| \sum_{1\leqslant {\;j}\leqslant t}c_j(r_j- s_j)\big|\leqslant 2^{\ell_0}c_0+\sum_{1\leqslant {\;j}\leqslant t}c_j.$
Therefore, when all
$r_j$
and all
$s_h$
are fixed except
$s_1$
, the number of admissible values for
$s_1$
is bounded. Summing (5·9) over
$r_j$
and
$s_h$
, taking the condition
$\sum_{1\leqslant {\;j}\leqslant t} 1/\ell_j=\delta$
into account then yields a contribution
\[\ll N^{\ell_0/\ell_1} x^{2\delta-1/\ell_1} {E_Q(x)\over \log x}\cdot\]
It remains to sum over
$\nu$
such that
$N^{\ell_0}=2^{\ell_0\nu}\leqslant 2^{\ell_0}x $
to get, in view of (5·7),
\[V_2^{\neq }(x;\,{\boldsymbol{c}},\,{\boldsymbol{\ell}})\ll x^{2\delta}{E_Q(x)\over \log x}\ll x^{2\delta}{{\mathfrak S}}(x).\]
Note that the manipulation resting on the condition
$m_0+n_0\in ]N,2N]$
saved a factor
$\log_2 x$
.
This completes the proof of Proposition 5·1.
5·4. Proof of Theorem 1·1.
Since
$V_1(x)\asymp x$
(see [
Reference Robert21
, lemma 3·1] for further precisions) and taking (1·5) into account the required estimate (1·3) immediately results from a suitable bound for
$V_2(x,{\boldsymbol{c}},{\boldsymbol{\ell}})$
. In view of (5·1), we write
${\boldsymbol{c}}=(c_1,\ldots,c_t),\,{\boldsymbol{\ell}}=(\ell_1,\ldots,\ell_t)$
.
For
$1\leqslant k\leqslant t$
, put
$\delta_k=\sum_{t-k\leqslant {\;j}\leqslant t} 1/\ell_j $
,
${\boldsymbol{c}}_k=(c_{t-k},\ldots, c_t)$
,
${\boldsymbol{\ell}}_k=(\ell_{t-k},\ldots, \ell_t)$
. The parameter s being defined in the statement of Theorem 1·1 and
${\boldsymbol{\ell}}$
satisfying conditions (i), (ii), (iii) of this statement, we shall show by induction on
$k\geqslant \max(1,s-1)$
that
\begin{equation}V_2(x;\,{\boldsymbol{c}}_k,{\boldsymbol{\ell}}_k)\ll x^{\delta_k}{{\mathfrak S}}(x) . \end{equation}
Note that hypothesis (iii) may be rewritten as
$ \delta_k\leqslant 1/\ell_{t-k-1}$
for
$s -1 \leqslant k\leqslant t -1. $
For
$k=s=1$
,
$t\geqslant 2$
, the stated bound (5·3) follows from [
Reference Robert21
, lemma 3·11]. The case
$k=1$
,
$s=2$
,
$t\geqslant 2$
, follows from (5·3). For
$k=2$
,
$t\geqslant 3$
,
$s=3$
, we only need to bound the number of solutions
$({\boldsymbol{x}},{\boldsymbol{y}})$
such that
$x_j, y_j\leqslant x^{1/\ell}$
and
\[x_1^\ell+x_2^\ell+x_3^\ell=y_1^\ell+y_2^\ell+y_3^\ell \]
with
$\ell=\ell_t$
. The required estimate then follows from Proposition 5·2. The induction is thus initialised. In these cases, the factor
${{\mathfrak S}}(x) $
is actually superfluous. Let
$k\geqslant s-1$
and assume (5·14) holds for k. We want to prove that it persists for
$k+1$
. By Propositions 5·2 and 5·1, we have
\[V_2(x ;\,{\boldsymbol{c}}_{k+1},\,{\boldsymbol{\ell}}_{k+1})\ll x^{ 2\delta_{k}}{{\mathfrak S}}(x) +x^{1/\ell_{t-k-1}} V_2(x ;\,{\boldsymbol{c}}_{k},{\boldsymbol{\ell}}_{k})\ll (x^{ 2\delta_{k} } +x^{1/\ell_{t-k-1}+\delta_k}){{\mathfrak S}}(x) .\]
This provides the desired bound since
$\delta_k\leqslant 1/\ell_{t-k-1}$
, and
$1/\ell_{t-k-1}+\delta_k= \delta_{k+1}$
.
The induction step is thus established, and so we get the result for
$k=t+1$
. This completes the proof of Theorem 1·1 since
$\delta_{t+1}=1$
.
Acknowledgements
The authors take pleasure in addressing warm thanks to Tim Browning, Carlo Pagano, Per Salberger, and the referee for pertinent remarks and suggestions.
