January 01, 1970
We study the Bateman–Horn conjecture for generalised von Mangoldt functions. As an application, we prove that for \(k \in \{2, 3\}\), almost all Bouniakowsky polynomials represent integers that are a product of exactly \(k\) primes.
Let \(k\) be a positive integer. A positive integer is called \(E_k\) when it has exactly \(k\) distinct prime divisors. A result of Chen [1] states that the polynomial \(x(x+2)\) represents infinitely many integers that are \(E_2\) or \(E_3\). Due to the parity problem it is not known which of the two cases occurs infinitely. Iwaniec [2] later proved that \(x^2+1\) takes on infinitely many values that are a prime or \(E_2\); again it is not known which case occurs infinitely often. The only irreducible integer polynomials that are known to represent infinitely many \(E_k\) are linear.
A polynomial \(P\in \mathbb{Z}[t]\) is called Bouniakowsky when it is irreducible, its leading coefficient is positive and for every prime \(\ell\) there is \(m\in \mathbb{Z}\) such that \(\ell\nmid P(m)\). Polynomials that represent infinitely many primes necessarily satisfy all these conditions and Bouniakowski conjectured that these are sufficient [3]. The analogous equivalent condition for representing infinitely many \(E_2\) numbers is not obvious. For example, \(P(t)=2t\) is not Bouniakowsky but still represents infinitely many such integers. However the density of \(E_2\) among the values of \(P(t)=2t\) is low and it is therefore more relevant to ask for a generalisation of the Bateman–Horn conjecture [4]. The conjecture states that for irreducible \(P\in \mathbb{Z}[t]\) one has \[\sum_{\substack{1\leqslant m\leqslant x\\P(m) \textrm{ prime } } } \log P(m) \sim \mathfrak S_{ P}x,\] where \(\mathfrak S_{P}\) is the singular series defined by \[\mathfrak S_{P}:=\prod_{\substack{ \ell \;\mathrm{ prime}\\ \ell=2 }}^\infty \frac{1-\ell^{-1} \sharp\{s\in \mathbb{F}_\ell:P(s)=0\}}{1-\ell^{-1}}.\] It seems that the right \(E_2\)-generalisation is as follows: an \(E_2\) integer \(m\) has the form \(m=p_1^{\alpha} p_2^{\beta}\) for primes \(p_1 \neq p_2\) and positive integers \(\alpha,\beta\). We then define \(\mathscr L_2(m)= (\log p_1)(\log p_2)\). For an irreducible \(P\in \mathbb{Z}[t]\) we ought to have \[\sum_{\substack{1\leqslant m\leqslant x\\P(m) \textrm{ is } E_2 } } \mathscr L_2(P(m)) \sim \frac{\mathfrak S_{ P}}{2} x (\log |P|x),\] where \(|P|\) is the maximum of absolute values of the coefficients of \(P\). Our first result proves this for \(100\%\) of polynomials ordered by height. We use the notation \(P>0\) to denote that the leading coefficient of \(P\) is strictly positive. We denote the version of \(\mathfrak S_{P}\) in which one takes the product over all primes \(\ell \leqslant\log x\) by \(\mathfrak S_{P}(x)\).
Theorem 1. Fix any \(d\in \mathbb{N}\) and \(\delta>1\). For any \(x,H\geqslant 1\) with \((\log H)(\log \log H)<x \leqslant(\log H)^\delta\) we have \[\sum_{\substack{ P\in \mathbb{Z}[t], \deg(P)=d \\ P>0, |P|\leqslant H}} \left|\sum_{\substack{1\leqslant m\leqslant x\\P(m) \;\mathrm{ is } \;E_2 } }\mathscr L_2(P(m)) -\frac{\mathfrak S_{ P}(x)}{2} x \log H\right|^2 \ll H^{d+1} \frac{(x\log H)^2}{\log x},\] where the implied constant depends only on \(\delta\) and \(d\). In particular, for \(100\%\) of degree \(d\) integer polynomials \(|P|\leqslant H\) with a positive leading coefficient one has \[\sum_{\substack{1\leqslant m\leqslant x\\P(m) \;\mathrm{ is }\; E_2 } }\mathscr L(P(m)) =\frac{\mathfrak S_{ P}(x)}{2} x \log H+O\Bigg( \frac{x\log H}{(\log x)^{1/4}} \Bigg)\] uniformly for \(x\leqslant(\log H)^\delta\). Furthermore, \(100\%\) of degree \(d\) Bouniakowsky polynomials \(P\) represent more than \((\log |P|)^\delta\) integers that are \(E_2\).
This generalises the analogous statement for primes in [5] Theorem 1.9. The proofs depend on expressing \(\mathscr
L_2\) through a linear combination of generalised von Mangoldt functions, which are defined by \[\Lambda_k(m):=
\sum_{\substack{ c\in \mathbb{N} \\ c\mid m }} \mu(c) \Big(\log \frac{m}{c} \Big)^k, \;\;\;k,m \in \mathbb{N}.\] These were initially investigated by Selberg for \(k=2\) in his proof of the Prime Number Theorem, and
subsequently studied for \(k \geqslant 3\) by Bombieri [6], [7]
and Friedlander–Iwaniec [8], [9], among others. The generalisation of Bateman–Horn’s conjecture
to \(\Lambda_k\) seems to be \[\sum_{\substack{1\leqslant m\leqslant x\\P_i(m)>0 \;\forall i} } \Lambda_{k_1}(P_1(m))\cdots \Lambda_{k_n}(P_n(m)) \sim
\mathfrak S_{\mathbf{P}} x
\prod_{i=1}^n k_i
(\log |P_i|x)^{k_i-1}
,\] where \(P_i\) are pairwise coprime irreducible integer polynomials and \[\mathfrak S_{\mathbf{P}} =
\prod_{\substack{ \ell \;\mathrm{ prime}\\ \ell=2 }}^\infty
\frac{1-\ell^{-1} \sharp\{s\in \mathbb{F}_\ell:P_1(s)\cdots P_n(s)=0\}}{(1-\ell^{-1})^n}.\] We prove this in \(100\%\) of the cases. Denote by \(\mathfrak S_{\mathbf{P}}(x)\) the
truncated version of \(\mathfrak S_{\mathbf{P}}\) in which the product is over primes \(\ell \leqslant\log x\). Fix \(n,d_1,\ldots, d_n \in \mathbb{N}\) and
let \[\textrm{Poly}(H):=\{\mathbf{P}\in (\mathbb{Z}[t])^n:
P_i>0,\deg(P_i)=d_i, |P_i|\leqslant H
\;\textrm{ for } i=1,\ldots, n\}.\]
The following is the main result in this paper:
Theorem 2. Fix any \(n\in \mathbb{N}, \mathbf{d}, \mathbf{k} \in \mathbb{N}^n,\delta>n\) and let \(d=\sum_{i=1}^n d_i, k=\sum_{i=1}^n k_i\). Then for any \(x,H\geqslant 1\) with \((\log H)^n<x \leqslant(\log H)^\delta\) we have \[\sum_{ \mathbf{P}\in \mathrm{Poly}(H)} \left| \sum_{\substack{1\leqslant m\leqslant x\\P_i(m)>0 \;\forall i} } \Lambda_{k_1}(P_1(m))\cdots \Lambda_{k_n}(P_n(m)) - \mathfrak S_{\mathbf{P}}(x) x (\log H)^{k-n} \prod_{i=1}^n k_i \right|^2 \! \! \! \ll \! \! \frac{H^{d+n} (x (\log H)^{k-n})^2}{\min\{ \log x,x(\log H)^{-n}\}},\] where the implied constant depends only on \(\delta,\mathbf{d},\mathbf{k}\) and \(n\).
Thus, for almost all \(P_1,\ldots, P_n\) one has \[\sum_{\substack{1\leqslant m\leqslant x\\P_i(m)>0 \;\forall i} } \Lambda_{k_1}(P_1(m))\cdots \Lambda_{k_n}(P_n(m)) \sim (k_1\cdots k_n) \mathfrak S_{\mathbf{P}}(x) x (\log H)^{k-n}.\] as soon as \(x/(\log H)^n\to \infty\).
Taking suitable combinations of \(\Lambda_k\) allows us to also handle \(E_3\) numbers. For a strictly positive \(E_3\) integer \(m\) let \(\mathscr L_3(m)\) be the product of the three distinct prime divisors of \(m\).
Theorem 3. Fix \(d\in \mathbb{N}\) and \(\delta>1\). For any \(x,H\geqslant 1\) with \((\log H)(\log \log H)<x \leqslant(\log H)^\delta\) we have \[\sum_{\substack{ P\in \mathbb{Z}[t], \deg(P)=d \\ P>0, |P|\leqslant H}} \left|\sum_{\substack{1\leqslant m\leqslant x\\P(m) \;\mathrm{ is } \; E_3 } }\mathscr L_3(P(m)) -\frac{\mathfrak S_{ P}}{6} x (\log H)^2\right|^2 \ll H^{d+1} \frac{(x\log H)^4}{\log x},\] where the implied constant depends only on \(\delta\) and \(d\). In particular, for \(100\%\) of degree \(d\) integer polynomials \(|P|\leqslant H\) with a positive leading coefficient one has \[\sum_{\substack{1\leqslant m\leqslant x\\P(m) \;\mathrm{ is } \; E_3 } }\mathscr L_3(P(m)) =\frac{\mathfrak S_{ P}}{6} x (\log H)^2+O\Bigg( \frac{x(\log H)^2}{(\log x)^{1/4}} \Bigg)\] uniformly for \(x\leqslant(\log H)^\delta\). Furthermore, \(100\%\) of degree \(d\) Bouniakowsky polynomials represent more than \((\log |P|)^\delta\) integers that are \(E_3\).
The analogous function \(\mathscr L_4\) for \(E_4\) numbers cannot be written as a combination of the \(\Lambda_k\) functions, see Lemma 11 for an exact statement. Crucially, \(\Lambda_k\) is supported on numbers of type \(E_j\) (\(j \leqslant k\)), but its average over \(E_j\) for any single \(j\leqslant k\) remains a positive proportion of the overall average.
Notation 1. Throughout this paper, \(\varphi\) denotes the Euler totient function, \(\mu\) denotes the Möbius function, and \(\ast\) represents the Dirichlet convolution. Additionally, we define the shorthand notation \(L(m) := \log m\). We let \(\omega(m)\) be the number of distinct prime numbers dividing \(m\).
We fix \(d,k,m_1\neq m_2\in \mathbb{N}\) and study \[\label{def:GmGm} \mathscr G_{(m_1,m_2)}(H;d,k):= \sum_{|P|\leqslant H} \Lambda_k(P( m_1)) \Lambda_k(P(m_2)),\tag{1}\] where the sum is over \(P\in \mathbb{Z}[t]\) of degree \(d\) with \(P>0\) and \(P(m_1),P(m_2)>0\).
Proposition 4. Fix \(\delta,\varepsilon>0\) and \(d,k\in \mathbb{N}\). For all integers \(1\leqslant m_1\neq m_2 \leqslant(\log H)^\delta\) we have \[\mathscr G_{\mathbf{m}}(H;d,k)=2^d H^{d+1} (k (\log H)^{k-1})^2 \frac{|m_1-m_2|}{\varphi(|m_1-m_2|)} \left(1+ O ( (\log H )^{- 1+ \varepsilon} )\right),\] where the implied constant is independent of \(m_i\) and \(H\).
The proof will be given in §2.2. For integers \(j_1,j_2\in [0,k]\) denote \[\begin{align} &\mathscr G_{(m_1,m_2)}(H;d,k)^{(j_1,j_2)}:=\sum_{ |P|\leqslant H } \prod_{h=1,2} (1\ast \mu L^{j_h})(P(m_h)) (\log P(m_1))^{k-j_h} \\ \;\textrm{ and } \; &\mathscr G^\flat_{(m_1,m_2)}(H;d,k)^{(j_1,j_2)}:= \sum_{ |P|\leqslant H } \prod_{h=1,2} (1\ast \mu L^{j_h})(P(m_h)) , \end{align}\] where the sums are over \(P\in \mathbb{Z}[t]\) of degree \(d\) with \(P>0\) and \(P(m_1),P(m_2)>1\).
Lemma 1 (Decomposing \(\Lambda_k\)). We have \[\mathscr G_{\mathbf{m}}(H;d,k)=\sum_{j_1,j_2=1 }^k {k\choose j_1 } {k\choose j_2 } (-1)^{j_1+j_2} \mathscr G_{\mathbf{m}}(H;d,k)^{\mathbf{j}},\]
Proof. By definition we have \(\Lambda_k(1) =0\). For \(m>1\) write \[\label{eq:pergolesi95euridice} \Lambda_k(m) = \sum_{j=0}^k {k\choose j } (\log m)^{k-j}(-1)^j (1\ast \mu L^j)(m)\tag{2}\] and note that \((1\ast \mu L^j)(m)=0\) vanishes for \(j=0\). ◻
We recall the bound \(0\leqslant\Lambda_k(m)\leqslant L(m)^k\) in [10]. Using this together with 2 and induction yields \[\label{eq:pergolesi95euridice95dove95sei} (1\ast \mu L^{k})(m) \ll_k L(m)^k ,\tag{3}\] with an implied constant independent of \(m\).
Lemma 2 (\(\log\) stabilisation). Keep the setting Proposition 4. For each \(j_1,j_2\in [1,k]\) we have \[\mathscr G_{\mathbf{m}}(H;d,k)^{\mathbf{j}}= (\log H)^{2k-j_1-j_2} \left( 1 +O\left( \frac{\log \log H}{\log H}\right)\right) \mathscr G^\flat_{\mathbf{m}}(H;d,k)^{\mathbf{j}} +O\left(\frac{H^{d+1}}{\log H}\right),\] where the implied constant is independent of \(m_i\) and \(H\).
Proof. Denote by \(E\) the subset of \(P\) appearing in \(\mathscr G_{\mathbf{m}}(H;d,k)^{\mathbf{j}}\) such that \(\max\{P(m_1),P(m_2)\}\) strictly exceeds \(H/(\log H)^{2k+1}\). We have \[\sharp E\leqslant\sum_{0<t\leqslant H/(\log H)^{2k+1}} \sharp\{|P|\leqslant H: (P(m_1)-t)(P(m_2)-t)=0\} \ll \frac{H}{(\log H)^{2k+1}}H^d,\] since the condition \((P(m_1)-t)(P(m_2)-t)=0\) fixes one coefficient of \(P\). The conditions of Proposition 4 ensure that \(\log P(m_i)\ll \log H\), thus, by 3 the contribution of \(E\) towards \(\mathscr G_{\mathbf{m} }(H;d,k)^{\mathbf{j}}\) is \(\ll \sharp E \cdot (\log H)^{2k} \ll H^{d+1}/\log H\).
When \(P\notin E\) we get \(\max\{P(m_1), P(m_2)\} > H^{d+1}/(\log H)^{2k+1}\), hence, \(\log (P(m_i)/H) \ll \log \log H\). In particular, for each \(i,j\) we obtain \((\log P(m_i))^j =(\log H)^j +O((\log H)^{j-1}(\log \log H))\), thus, \[\mathscr G_{\mathbf{m}}(H;d,k)^{\mathbf{j}}= O\left(\frac{H^{d+1}}{\log H}\right)+ (\log H)^{2k-j_1-j_2} \left( 1 +O\left( \frac{\log \log H}{\log H}\right)\right) \sum_{ \substack{ |P|\leqslant H } } \prod_{h=1,2} (1\ast \mu L^{j_h})(P(m_h)) ,\] where the sum over \(P\) in the right-hand side is subject to \(\max\{P(m_i):i=1,2\}> H/(\log H)^{2k+1}\) and the conditions present in \(\mathscr G^\flat_{\mathbf{m}}(H;d,k)^{\mathbf{j}}\). Ignoring the condition involving \(\max\{P(m_i):i=1,2\}\) introduces an error term of size \(\ll (\log H)^{2k-j_1-j_2} \sharp E \cdot (\log H)^{j_1+j_2} \ll H^{d+1}/\log H\) by 3 . ◻
Fix \(j\geqslant 1\), \(z\geqslant 2\) and for \(m\in \mathbb{N}\) define \[\mathscr E_{z,j}(r):=\sum_{\substack{c\in \mathbb{N}, c\mid m \\ c>z } } \mu(c) (\log c)^j.\] This generalizes the function \(\mathscr E_{z}=\mathscr E_{z,1}\) in [5].
Lemma 3 (Exponential sum input). Fix \(A > 0\). Then for all \(j\geqslant 1\) and \(1\leqslant z \leqslant y\) we have \[\sup_{\alpha \in \mathbb{R}} \left| \sum_{m \in \mathbb{N} \cap [1, y]} \mathscr{E}_{z,j}(m) \mathrm{e}^{i r\alpha} \right| \ll \frac{ y \log y}{(\log z)^{A}},\] where the implied constant depends only on \(A\) and \(j\).
Proof. We write the sum as \[\sum_{\substack{ c t \leqslant y \\ c>z }} \mu(c) (\log c)^j \mathrm{e}^{i ct \alpha} = \sum_{ t \leqslant y/z } \sum_{ z< c \leqslant y/t } \mu(c) (\log c)^j \mathrm{e}^{i ct \alpha} .\] By the theorem of Davenport [11] and partial summation, the inner sum is \(O((y/t)(\log (y/t))^{-A} ))\), hence, we obtain the overall bound \[\ll \sum_{ t \leqslant y/z } \frac{y}{t (\log (y/t))^{A}} \leqslant\frac{y }{(\log z)^{A}} \sum_{ t \leqslant y } \frac{1}{t}\ll \frac{y \log y}{(\log z)^{A}} .\qedhere\] ◻
The following lemma is an analogue of [5]; it can be proved by injecting Lemma 3 into [5]. Denote by \[\mathscr G^{\flat\flat}_{(m_1,m_2)}(H;d,k)^{(j_1,j_2)}\] the expression defined by \(\mathscr G^\flat_{(m_1,m_2)}(H;d,k)^{(j_1,j_2)}\) when \(1\ast \mu L^{j_h}\) is replaced by \(1\ast ( \mu L^{j_h}\mathbb{1}_{[1,z]})\).
Lemma 4 (Möbius randomness). Keep the setting Proposition 4 and fix \(j_1,j_2\in [1,k],A>0\) and \(\eta\in (0,1)\). Then for all \(z, H \geqslant 1\) such that \(\exp((\log H)^\eta) \leqslant z \leqslant H\) we have \[\mathscr G^\flat_{\mathbf{m}}(H;d,k)^{\mathbf{j}}= \mathscr G^{\flat\flat}_{\mathbf{m}}(H;d,k)^{\mathbf{j}}+ O \left( \frac{H^{d+1}}{(\log H)^A} \right),\] where the implied constant does not depend on \(m_1, m_2, H\) and \(z\).
The only difference compared to [5] is that the condition \(z\geqslant H^{\delta_1}\) is relaxed to \(\exp((\log H)^\eta) \leqslant z\), something that was also possible in [5] but not recorded. This alternative choice of \(z\) can be explained by noting that Lemma 3 provides the succeeding error term upon taking \(y=H (\log H)^{\delta}\) and \(z=\exp((\log H)^\eta)\): \[\frac{y \log y }{ (\log z)^{(A+1+\delta)/\eta} } = \frac{H \log (H (\log H)^\delta) }{( (\log H)^{\eta})^{(A+1+\delta)/\eta} } \ll \frac{H \log H }{ (\log H)^A }.\] This different choice for \(z\) allows us to give a weak upper bound for \(\mathscr G_{\mathbf{m}}(H;d,k)^{\mathbf{j}}\) but one that is sufficient for our purposes.
Lemma 5. Keep the setting Lemma 4and take \(z=\exp((\log H)^\eta)\). Then we have \[\mathscr G^{\flat\flat}_{\mathbf{m}}(H;d,k)^{\mathbf{j}} \ll H^{d+1} (\log H )^{\eta( j_1+j_2+2)}(\log \log H),\]where the implied constant is independent from \(m_i\) and \(H\).
Proof. We can upper-bound the sum by \[\sum_{c_1,c_2 \leqslant z} |\mu(c_1)\mu(c_2)| (\log c_1)^{j_1}(\log c_2)^{j_2} \sharp\{ P\in \mathbb{Z}[t]: \deg(P)=d, |P|\leqslant H, c_i \mid P(m_i), i=1,2 \} .\] Our choice of \(z\) makes apparent that \((\log c_i)^{j_i} \leqslant(\log z)^{j_i} =(\log H)^{\eta j_i}\). The cardinality of \(P\) was studied in the arguments appearing in [5]. Writing \(l_0= \gcd(c_0,c_1)\) and \(l_i=c_i/l_0\) for \(i=1,2\) and using the estimates proved in [5] we obtain the bound\[\sharp\{ P\in \mathbb{Z}[t]: \deg(P)=d, |P|\leqslant H, c_i \mid P(m_i), i=1,2 \} \ll H^{d+1} \frac{\gcd(l_0,l_1)}{l_0^2 l_1 l_2 }.\] This gives the overall estimate \[\ll H^{d+1} (\log H)^{\eta (j_1+j_2)} \sum_{\substack{ l_0,l_1,l_2 \in \mathbb{N} \\ l_1,l_2 \leqslant z}} |\mu(l_0 )| \frac{\gcd(l_0,m_1-m_2) }{l_0^2 l_1 l_2 } \ll H^{d+1} (\log H)^{\eta (j_1+j_2)} \prod_{p\mid m_1-m_2} \left(1+\frac{1}{p}\right) .\] Since \(|m_1-m_2| \leqslant(\log H)^\delta\) we may use Mertens’ theorem to see that\[\prod_{p\mid m_1-m_2} \left(1+\frac{1}{p}\right) \ll \log |m_1-m_2| \ll \log \log H.\qedhere\] ◻
Lemma 6. Fix \(\varepsilon>0\) and \(j_1,j_2 \in [1,k]\). In the setting of Proposition 4 we have \[\mathscr G_{\mathbf{m}}(H;d,k) =k^2 (\log H)^{2k-2} \mathscr G^\flat_{\mathbf{m}}(H;d,k)^{(1,1)} \left( 1 +O\left( \frac{\log \log H}{\log H}\right)\right) + O( H^{d+1} (\log H )^{2k- 3+\varepsilon}) ,\] where the implied constant is independent from \(m_i\) and \(H\).
Proof. Injecting Lemma 5 with \(\eta=\varepsilon/(4k+4)\) into Lemma 4 shows that \[\mathscr G^\flat_{\mathbf{m}}(H;d,k)^{\mathbf{j}} \ll H^{d+1} (\log H )^{\varepsilon/2} (\log \log H)\ll H^{d+1} (\log H )^\varepsilon.\] Feeding this into Lemma 2 we then obtain \[\mathscr G_{\mathbf{m}}(H;d,k)^{\mathbf{j}} \ll H^{d+1} (\log H )^{2k- j_1-j_2+\varepsilon} .\] We may then use this for all pairs \((j_1,j_2)\neq (1,1)\) in Lemma 1 to obtain \[\mathscr G_{(m_1,m_2)}(H;d,k)= k^2 \mathscr G_{(m_1,m_2)}(H;d,k)^{(1,1)}+ O\left( H^{d+1} (\log H )^{2k- 3+\varepsilon} \right).\] Finally, alluding to Lemma 2 with \(j_1=j_2=1\) shows that \(\mathscr G_{(m_1,m_2)}(H;d,k)\) equals \[k^2 (\log H)^{2k-2} \mathscr G^\flat_{\mathbf{m}}(H;d,k)^{(1,1)} \left( 1 +O\left( \frac{\log \log H}{\log H}\right)\right) + O( H^{d+1} (\log H )^{2k- 3+\varepsilon} ),\] which concludes the proof. ◻
The main term in Lemma 6 contains \(\mathscr G^\flat_{\mathbf{m}}(H;d,k)^{(1,1)}\), which coincides with the sum in [5]. Hence, by [5] one has \[\mathscr G^\flat_{\mathbf{m}}(H;d,k)^{(1,1)}= 2^d H^{d+1} \frac{|m_1-m_2|}{\varphi(|m_1-m_2|)} +O\left( \frac{ H^{d+1}}{ \log H}\right).\] Combining this with Lemma 6 we obtain \[\mathscr G_{\mathbf{m}}(H;d,k) =2^d k^2 H^{d+1} (\log H)^{2k-2} \frac{|m_1-m_2|}{\varphi(|m_1-m_2|)} + O ( H^{d+1} (\log H )^{2k- 3+3\varepsilon} ),\] where we used the bound \[\frac{|m_1-m_2|}{\varphi(|m_1-m_2|)} \ll \prod_{p\leqslant|m_1-m_2|} (1-p^{-1})^{-1} \ll \log |m_1-m_2| \ll \log x \ll \log \log H\] by Mertens’ theorem. This completes the proof of Proposition 4.
Opening up the second moment in the left-hand side of Theorem 2 we get \[\label{eq:karotsaki} V_1-2k_1\cdots k_n x (\log H)^{k-n} V_2 +(k_1\cdots k_n x (\log H)^{k-n})^2 V_3 ,\tag{4}\] where \[\begin{align} V_1& := \sum_{1\leqslant m_1,m_2\leqslant x } \prod_{i=1}^n \sum_{\substack{P \in \mathbb{Z}[t], \deg(P)=d_i,P>0 \\|P|\leqslant H, P(m_1),P(m_2)>0 }} \Lambda_{k_i}(P(m_1)) \Lambda_{k_i}(P(m_2)), \\ V_2& := \sum_{1\leqslant m\leqslant x } \sum_{\substack{\mathbf{P} \in \textrm{Poly}(H) \\ P_i(m)>0 \forall i }} \mathfrak S_{\mathbf{P} }(x) \prod_{i=1}^n \Lambda_{k_i}(P_i(m)) , \\ V_3& := \sum_{\mathbf{P} \in \textrm{Poly}(H)} \mathfrak S_{\mathbf{P} }(x)^2. \end{align}\] Let us recall that [5] states that for \(x \leqslant H^{1/4}\) we have \[\label{lem:ch39io32muti32consiglio344432Guglielmo}V_3 = 2^d H^{d+n} \prod_{\substack{\mathrm{prime } \;\ell \\ \ell \leqslant\log x}} \gamma_n(\ell) + O(H^{d+n-1/2}),\tag{5}\] where \[\gamma_n(\ell) := 1 - \frac{1}{\ell} + \frac{\ell^{n-1}}{(\ell - 1)^n}=1+\frac{n}{\ell^2} +O_n\left( \frac{1}{\ell^3} \right)\] and the implied constant is independent of \(x\) and \(H\).
We split \(V_1\) as \(V_1= V_1^{\textrm{diag}}+V_1^{\textrm{non-diag}},\) where \(V_1^{\textrm{diag}}\) denotes the contribution of the terms \(m_1=m_2\) towards \(V_1\). Recalling 1 we apply Proposition 4 to write \(V_1^{\textrm{non-diag}}\) as \[\begin{align} &\sum_{1\leqslant m_1\neq m_2\leqslant x } \prod_{i=1}^n \mathscr G_{\mathbf{m}}(H;d_i,k_i)\\ &=\left(1+ O ( (\log H )^{- 1+ \varepsilon} )\right) 2^d H^{d+n} (\log H)^{2k-2n} k_1^2\cdots k_n^2 \sum_{1\leqslant m_1\neq m_2\leqslant x } \Big( \frac{|m_1-m_2|}{\varphi(|m_1-m_2|)} \Big)^n . \end{align}\] The sum over \(m_1 \neq m_2\) in the last display coincides with the quantity \(2T_0(x)\) (when \(M=1\)) in the proof of [5]. In that lemma it was proved that \[\sum_{1\leqslant m_1< m_2\leqslant x } \Big( \frac{|m_1-m_2|}{\varphi(|m_1-m_2|)} \Big)^n= \frac{x^2}{2} \prod_{\substack{\mathrm{prime } \;\ell \\ \ell =2 }}^\infty \gamma_n(\ell) +O(x^{3/2}).\] Truncating the product in the right-hand side to primes \(\ell \leqslant\log x\) in order to get something closer to 5 we introduce an error term of size \(O(x^2/\log x)\) due to \(\gamma_n(\ell)=1+n \ell^{-2} +O_n(\ell^{-3})\). Thus, \[\label{Dunque32si32sfoga}V_1^{\textrm{non-diag}} = 2^d H^{d+n} (\log H)^{2k-2n} k_1^2\cdots k_n^2 x^2 \prod_{\substack{\mathrm{prime } \;\ell \\ \ell \leqslant\log x}} \gamma_n(\ell)\left(1+ O ( (\log x)^{- 1 } )\right).\tag{6}\] Before proceeding to the estimation of \(V_1^{\textrm{diag}}\) let us note the following:
Remark 5. One can use the bound \(\Lambda_k\leqslant L^k\) to give a crude bound for \(V_1^{\textrm{diag}}\), however, this would eventually result in a weaker version of Theorem 2, in which the asymptotic holds in a smaller range for \(x\). In light of \[\label{Lo32so32ma32il32mio32timore} M (\log M)^{2k-1} \ll \sum_{m\leqslant M } \Lambda_k(m) ^2\ll M (\log M)^{2k-1}\tag{7}\] we expect that \[\label{Misero3332In32quale32abisso} V_1^{\textrm{diag}}\ll x H^{d+n} (\log H)^{2k-n}.\tag{8}\] To justify 7 we use \(\Lambda_{k+1}=\Lambda_k L+ \Lambda_k \ast \Lambda\) from [10] in order to get \(\Lambda_{k+1}\geqslant\Lambda_k L\) and by induction, \(\Lambda_{k}\geqslant\Lambda_1 L^k=\Lambda L^{k-1}\). Hence, \[\sum_{m\leqslant M } \Lambda_k(m) ^2 \gg (\log M)^{2k-2} \sum_{m \in (M/2,M] } \Lambda(m) ^2 \gg M (\log M)^{2k-1}\] by restricting to primes \(m\) and using the Prime Number Theorem. For the upper bound we use \(\Lambda_k \leqslant L^k\) from [10] to get \[\sum_{m\leqslant M } \Lambda_k(m) ^2\leqslant(\log M)^k\sum_{m\leqslant M } \Lambda_k(m) \ll M (\log M)^{2k-1}\] by [10].
Let us now continue by verifying 8 when \(x\geqslant(\log H)^n\). By \(\Lambda_k \leqslant L^k\) we obtain \[\label{Rosique} V_1^{\textrm{diag}} = \sum_{1\leqslant m\leqslant x } \sum_{\substack{\mathbf{P} \in \textrm{Poly}(H)\\P_i(m)>0\forall i }} \prod_{i=1}^n \Lambda_{k_i}(P_i(m))^2 \ll (\log H)^{k} \sum_{\mathbf{P} \in \textrm{Poly}(H)} \psi_{\mathbf{P}}(x)=: (\log H)^{k} Z, \;\textrm{ say},\tag{9}\] where \[\psi_{\mathbf{P}}(x):= \sum_{\substack{m\leqslant x\\P_i(m)>0\forall i }} \prod_{i=1}^n \Lambda_{k_i}(P_i(m)).\] It remains to prove \(Z\ll H^{d+n} x(\log H)^{k-n}\). By Cauchy’s inequality we obtain \[\begin{align} Z^2 &\ll H^{d+n} \sum_{\mathbf{P} \in \textrm{Poly}(H)} \psi_{\mathbf{P} }(x)^2 =H^{d+n} (V_1^{\textrm{diag}}+V_1^{\textrm{non-diag}})\\ &\ll H^{d+n} ( (\log H)^k Z + H^{d+n} (\log H)^{2k-2n} x^2) \end{align}\] by 6 and 9 . If the right hand side is dominated by the second term then \[Z^2 \ll (H^{d+n} (\log H)^{k-n} x)^2,\] which is acceptable. If the first term dominates then by \(x\geqslant(\log H)^n\) we obtain \[Z \ll H^{d+n} (\log H)^{k} = H^{d+n} (\log H)^{k- n} (\log H)^n \leqslant H^{d+n} (\log H)^{k- n} x,\] which is satisfactory. Bringing together 6 and 8 shows that when \(x\geqslant(\log H)^n\) then \[\label{finalv1} V_1= 2^d H^{d+n} (\log H)^{2k-2n} k_1^2\cdots k_n^2 x^2 \prod_{ \ell \leqslant\log x} \gamma_n(\ell) +O\Big( \frac{ H^{d+n} x^2}{ (\log H)^{2n-2k} } \Big( \frac{(\log H)^n}{x} + \frac{1}{\log x} \Big) \Big).\tag{10}\] To estimate the mixed term \(V_2\), we apply Knafo’s generalization of the Siegel–Walfisz theorem for \(\Lambda_k\) [12], specializing it to the range \(T =\exp(\sqrt{ \log x})\) and \(q \leqslant(\log y)^A\). Following [12], there exist explicit constants \(a_n(q)\) that satisfy \(a_0(q)=1\) and the bound \(a_n(q)\ll_n (\log \log q)^n\) by [12], yielding the following result:
Lemma 7 (Knafo). Fix \(k\geqslant 1\) and \(A,B>0\). Then for all \(y,q\geqslant 1\) with \(q\leqslant(\log y)^A\) and all \(b\in (\mathbb{Z}/q\mathbb{Z})^*\) we have \[\sum_{\substack{1\leqslant m \leqslant y \\ m\equiv b \left(\textrm{mod}\ q\right) }} \Lambda_k(m) = \frac{ y }{\varphi(q)}\Big[\sum_{j=1}^k {k \choose j } a_{j-1}(q)(\log y)^{k-j}\Big]+ O\Big(\frac{y}{(\log y)^B} \Big) ,\] where the implied constant depends only on \(A,B\) and \(k\).
Corollary 1. Fix \(k\geqslant 1\) and \(A,B,C>0\). Then for all \(y,q\geqslant 1\) with \(q\leqslant(\log y)^A\), all \(b\in (\mathbb{Z}/q\mathbb{Z})^*\) and all \(y (\log y)^{-C}\leqslant Y \leqslant y (\log y)^C\) we have \[\sum_{\substack{m \in [Y,Y+y] \\ q \mid m-b }} \Lambda_k(m)= \frac{ ky }{\varphi(q)} (\log y)^{k-1} + O\Big( \frac{ y(\log y)^{k-7/4}}{\varphi(q)}\Big) = \frac{ ky (\log y)^{k-1} }{\varphi(q)} (1 + O( (\log y)^{-3/4} ) ),\] where the implied constant depends only on \(A,B,C\) and \(k\).
Proof. By Lemma 7 with \(B=A+C+1\) we obtain \[\label{sospirat} \frac{1}{\varphi(q)}\sum_{j=1}^k {k \choose j } a_{j-1}(q) \int_Y^{y+Y} ( t (\log t)^{k-j} )' \mathrm d t + O\Big(\frac{Y}{\varphi(q)(\log y)^{1+C}} \frac{\varphi(q)}{(\log y )^A} \Big).\tag{11}\] Since \(Y\leqslant y (\log y)^C\) and \(\varphi(q) \leqslant q \leqslant(\log y )^A\) the error term is satisfactory. For a constant \(\omega \geqslant 0\) we have \[\int_Y^{y+Y} ( t (\log t)^{\omega} )' \mathrm d t =\int_Y^{y+Y} (\log t)^{\omega-1} (\omega + \log t) \mathrm d t \ll_\omega y (\log y)^{\omega} ,\] hence, the contribution of \(j\neq 1\) towards 11 is \[\ll_k \frac{y}{\varphi(q)}\sum_{j=2}^k a_{j-1}(q) (\log y)^{k-j} \ll \frac{y}{\varphi(q)} \sum_{j=2}^k (\log \log q ) ^{j-1} (\log y)^{k-j} \ll \frac{y}{\varphi(q)}(\log \log q)^k (\log y)^{k-2},\] which is acceptable. Similarly, \[\int_Y^{y+Y} ( t (\log t)^{\omega} )' \mathrm d t =\int_Y^{y+Y} (\log t)^{\omega-1} (\omega + \log t) \mathrm d t =\int_Y^{y+Y}(\log t)^\omega \mathrm d t+O( y (\log y)^{\omega -1 } )\] and using the fact that both \(\log Y\) and \(\log (y+Y)\) are \(\log y + O(\log \log y )\) we find that \[\int_Y^{y+Y}(\log t)^\omega \mathrm d t = (\log y )^\omega \int_Y^{y+Y}(1+O((\log \log t)/\log t)) \mathrm d t = y (\log y )^\omega \Big( 1+ O\Big ( \frac{ \log \log y }{\log y } \Big) \Big ).\] Applying this with \(\omega=k-1\) to the term \(j=1\) in 11 completes the proof. ◻
To bound \(V_2\), let \(W\) be the product of all primes \(\ell \leqslant\log x\). Arguing as in [5], it follows that \[V_2=\sum_{1\leqslant m\leqslant x } \sum_{\substack{ \mathbf{R} \in (\mathbb{Z}/W)[t]^n\\ \deg(R_i)\leqslant d_i \forall i }} \mathfrak S_{\mathbf{R} }(x) \prod_{i=1}^n \sum_{\substack{P \in \mathbb{Z}[t],|P|\leqslant H \\ P_i \equiv R_i \mod W }} \Lambda_{k_i}(P(m)) ,\] where the sum over \(P\) is subject to \(\deg(P)=d_i, P>0\) and \(P(m)>0\). Since both \(\Lambda_{k_i} , \mathfrak S_{\mathbf{R} }(x)\) are non-negative we may lower bound \(V_2\) by \[\label{lowerbound} V_2\geqslant\sum_{ m\leqslant x } \sum_{\substack{ \mathbf{R} \in (\mathbb{Z}/W)[t]^n\\ \gcd(R_i(m),W)=1, \deg(R_i)\leqslant d_i \forall i }} \mathfrak S_{\mathbf{R} }(x) \prod_{i=1}^n \sum_{\substack{P \in \mathbb{Z}[t],|P|\leqslant H \\ P_i \equiv R_i \mod W }} \Lambda_{k_i}(P(m)).\tag{12}\] Letting \(P(t)=\sum_{j=0}^d c_j m^j, N=P(m)-c_0\) and \(t=P(m)\) we can write the sum over \(P\) as \[\label{lowerbound23} \sum_{c_1,\ldots c_d} \sum_{\substack{t>0,|t-N|\leqslant H\\ t \equiv R_i(m) \mod W }} \Lambda_{k_i}(t ),\tag{13}\] where the sum over \(c_1,\ldots, c_d\) is subject to \(|c_j|\leqslant H\), \(c_d>0\) and \(c_j\) being congruent modulo \(W\) to the \(j\)-th coefficients of \(R_i\). We have \[\log W =\sum_{\ell\leqslant\log x }\log \ell \ll \log x < (\log H)^\delta,\] hence \(W\leqslant(\log H)^{O(1)}\), which allows us to employ Corollary 1 with \(q=W,Y=N\) and \(y=H\) to deal with the range \(t \in [N,N+H]\). Applying the same corollary again to deal with the range \(t \in [N-H,N]\) yields the overall estimate \[\sum_{\substack{t>0,|t-N|\leqslant H\\ t \equiv R_i(m) \mod W }} \Lambda_{k_i}(t )= \frac{2k_i H}{\varphi(W)} (\log H)^{k_i-1} (1+O( (\log H)^{-3/4})).\] Injecting this into 13 yields \[\sum_{c_1,\ldots c_d} \sum_{\substack{t>0,|t-N|\leqslant H \\ t \equiv R_i(m) \mod W }} \Lambda_{k_i}(t ) =2^{d_i} k_i \frac{H^{1+d_i}}{W^{d_i}} (\log H)^{k_i-1} (1+O( (\log H)^{-3/4})).\] This estimate can in turn be fed into 12 resulting in \[V_2\geqslant 2^d (k_1\cdots k_n) \frac{H^{d+n}}{W^d \varphi(W)^n} ( \log H)^{k-n} (1+O( (\log H)^{-3/4}))\sum_{ m\leqslant x } \sum_{\mathbf{R} \in (\mathbb{Z}/W)[t]^n } \mathfrak S_{\mathbf{R} }(x)\] where the sum over \(\mathbf{R}\) is subject to \(\gcd(R_i(m),W)=1\) and \(\deg(R_i)\leqslant d_i\) for all \(i\). By alluding to [5] we can estimate the inner sum over \(\mathbf{R}\) as \[\varphi(W)^n W^d \prod_{\ell \leqslant\log x } \gamma_n(\ell),\] which is independent of \(m\). Summing over \(m\) yields \[V_2\geqslant 2^d (k_1\cdots k_n) x H^{d+n} ( \log H)^{k-n} \Big(\prod_{\ell \leqslant\log x } \gamma_n(\ell) \Big) (1+O( x^{-1}+(\log H)^{-3/4})) .\] Injecting this estimate together with 5 and 10 to 4 completes the proof of Theorem 2.
Lemma 8. Fix \(d\in \mathbb{N}\) and \(A>0\). Then the number of \(P\in \mathbb{Z}[t]\) with \(\deg(P)=d,|P|\leqslant H\) such that there exists \(m\in [1,x]\) with \(|P(m)|\leqslant H (\log H)^{-A}\) is \(\ll x H^{d+1} (\log H)^{-A}\), with an absolute constant that is independent of \(x\) and \(H\).
Proof. By the union bound we get \[\leqslant\sum_{m\leqslant x} \sum_{\substack{ (c_1,\ldots, c_d) \in \mathbb{Z}^d\cap[-H,H]^d\\ |t| \leqslant H (\log H)^{-A} }}\sharp\left\{|c_0|\leqslant H: \sum_{j=0}^d c_j m^j =t\right\} \leqslant\sum_{m\leqslant x} \sum_{\substack{ (c_1,\ldots, c_d) \in \mathbb{Z}^d\cap[-H,H]^d\\ |t| \leqslant H (\log H)^{-A} }} 1 ,\] which is \(\ll xH^{d+1}/(\log H)^A\). ◻
Lemma 9. Fix \(d\in \mathbb{N}\) and \(\delta,j,c_1,c_2,c_3>0\) and let \(\mathscr C_P\in \mathbb{C}\) be a constant defined for each \(P\in \mathbb{Z}[t]\) such that \(|\mathscr C_P| \leqslant(\log |P|)^{c_3}\) for all \(P\). Let \(f:\mathbb{N} \to \mathbb{C}\) be such that \(|f(m)| \leqslant c_1 (\log (2+m))^{c_2}\) for all \(m\). Then for all \(x\leqslant(\log H)^\delta\) the quantity \[\sum_{\substack{ P\in \mathbb{Z}[t],|P|\leqslant H \\ \deg(P)=d}} \left( \sum_{\substack{ m\leqslant x\\P(m)>0} } f(P(m)) (\log P(m))^j - \mathscr C_P (\log H)^j\right)^2\]equals \[(\log H)^{2j}\Big( 1 +O\Big(\frac{\log \log H}{\log H} \Big)\Big) \sum_{\substack{ P\in \mathbb{Z}[t],|P|\leqslant H \\ \deg(P)=d}} \left( \sum_{\substack{ m\leqslant x\\P(m)>0} } f(P(m)) - \mathscr C_P \right)^2 +O(H^{d+1} (\log H)^{-A} )\]where the implied constant depends at most on \(A,d,j\) and \(c_i\).
Proof. Our assumptions imply that \[\Big| \sum_{\substack{ m\leqslant x\\P(m)>0} } f(P(m)) (\log P(m))^j - \mathscr C_P (\log H)^j\Big|\ll (x(\log H)^{c_2}+ (\log H)^{c_3}) (\log H)^{j}\ll (\log H)^{lA_0},\] with \(A_0=j+\max\{ \delta+c_2,c_3\}\). Thus, the contribution of \(P\) in Lemma 8 is \[\ll xH^{d+1} (\log H)^{-A+2A_0}\ll H^{d+1} (\log H)^{\delta-A+2A_0}\leqslant H^{d+1} (\log H)^{-A/2}\] when \(A>2(\delta+4A_0)\). As in the proof of Lemma 2 we see that for the remaining \(P\) and all \(m\) in \([1,x]\) we have \((\log P(m))^j =(\log H)^j +O((\log H)^{j-1}(\log \log H))\). Hence, up to an admissible error term, the quantity in the lemma equals \[(\log H)^{2j} \Big( 1 +O\Big(\frac{\log \log H}{\log H} \Big)\Big) \mathop{\mathrm{\sum{}^*}}_{\substack{ P\in \mathbb{Z}[t],|P|\leqslant H \\ \deg(P)=d}} \left( \sum_{\substack{ m\leqslant x\\P(m)>0} } f(P(m)) - \mathscr C_P \right)^2,\] where \(\mathop{\mathrm{\sum{}^*}}\) is the sum over all \(P\) not appearing in Lemma 8. We can now complete the sum over \(P\) by using once again Lemma 8. ◻
Lemma 10. Fix \(d,k\in \mathbb{N}\) and \(\delta,j>0\). Then for all \(x\leqslant(\log H)^\delta\) the quantity \[\sum_{\substack{ P\in \mathbb{Z}[t],|P|\leqslant H \\ \deg(P)=d}} \left( \sum_{\substack{ m\leqslant x\\P(m)>0} } \Lambda_k(P(m)) (\log P(m))^j - \mathfrak S_P(x) (\log H)^j\right)^2\]equals \[(\log H)^{2j}\Big( 1 +O\Big(\frac{\log \log H}{\log H} \Big)\Big) \sum_{\substack{ P\in \mathbb{Z}[t],|P|\leqslant H \\ \deg(P)=d}} \left( \sum_{\substack{ m\leqslant x\\P(m)>0} } \Lambda_k(P(m)) -\mathfrak S_P(x) \right)^2 +O(H^{d+1} (\log H)^{-A} )\]where the implied constant depends only on \(A,d\) and \(j\).
Proof. The proof follows directly from Lemma 8 by recalling that \(0\leqslant\Lambda_k \leqslant L^k\) and noting that \[\mathfrak S_ P(x) \leqslant\prod_{\ell \leqslant\log x } (1-1/\ell)^{-1} \ll \log \log x.\qedhere\] ◻
Proof. The proof starts from [10], which states\[\Lambda_{k+1}=\Lambda_k L+ \Lambda_k \ast \Lambda.\] In particular, \(\Lambda \ast \Lambda=- \Lambda_{2}+\Lambda L\), thus, we handle \(\Lambda L\) with Lemma 10 with \(j=k=1\) and \(- \Lambda_{2}\) with Theorem 2 with \(n=1,k_1=2\) to obtain \[\label{Abel17} \sum_{\substack{ P\in \mathbb{Z}[t],|P|\leqslant H \\ \deg(P)=d}} \left| \sum_{\substack{1\leqslant m\leqslant x\\P(m)>0 } } (\Lambda \ast \Lambda)(P(m)) - \mathfrak S_{P}(x) x \log H \right|^2 \ll \frac{H^{d+1} (x \log H)^2}{\log x}\tag{14}\] for \((\log H)(\log \log H) \leqslant x \leqslant(\log H)^\delta\).
For a prime \(p\) and a positive integer \(\alpha\) the quantity \((\Lambda\ast \Lambda)(p^\alpha)\) vanishes when \(\alpha=1\) and is \(O((\log (p^\alpha))^2)\) otherwise. Thus, the contribution of \(m,P\) such that \(P(m)\) is a power of a prime is \[\sum_{\substack{ P\in \mathbb{Z}[t],|P|\leqslant H \\ \deg(P)=d}} \left| \sum_{\substack{1\leqslant m\leqslant x\\P(m)=p^\alpha } } (\Lambda \ast \Lambda)(P(m)) \right|^2 \ll x (\log H)^2 \sum_{m\leqslant x} \sum_{\substack{ P\in \mathbb{Z}[t],|P|\leqslant H \\ \deg(P)=d,P(m)=p^\alpha } } (\Lambda \ast \Lambda)(P(m)).\] This is \(\ll H^{d+1/2+\varepsilon}\) for any fixed \(\varepsilon>0\) since \((\Lambda \ast \Lambda)(P(m))=O(H^\varepsilon)\), \(x\leqslant(\log H)^\delta\) and \[\begin{align} &\sharp\{ P\in \mathbb{Z}[t],|P|\leqslant H:\deg(P)=d,P(m)=p^\alpha, \alpha =2\} \\ \leqslant&\sum_{2\leqslant\alpha \ll \log H} \sum_{p \ll (x^d H)^{1/\alpha}} \sharp\{ P\in \mathbb{Z}[t],|P|\leqslant H: \deg(P)=d,P(m)=t\}\ll (\log H) (x^d H)^{1/2} H^d \end{align}\] uniformly in \(m\). Since \(\Lambda\ast \Lambda\) is supported on integers \(m\) with \(\omega(m)\leqslant 2\) we deduce \[\sum_{\substack{ P\in \mathbb{Z}[t],|P|\leqslant H \\ \deg(P)=d}} \left| \sum_{\substack{ m\leqslant x, P(m)>0\\ P(m) \textrm{ is } E_2 } } (\Lambda \ast \Lambda)(P(m)) - \mathfrak S_{P}(x) x \log H \right|^2 \ll \frac{H^{d+1} (x \log H)^2}{\log x}\] when \(x\leqslant\log H\). To finish the proof of the first claim of Theorem 1 we use the fact that \[\omega(t)=2 \Rightarrow (\Lambda\ast \Lambda)(t)=2 \prod_{ \textrm{ prime } \ell \mid t} \log \ell =2 \mathscr L_2(t)\]for \(t=P(m)\). The second claim of Theorem 1 follows from the first and Chebychev’s inequality. The third claim of Theorem 1 follows from the second and [2], which shows that \(100\%\) of Bouniakowsky polynomials \(P\) satisfy \(\mathfrak S_P(x)\gg (\log \log x)^{1-\deg(P)}\). ◻
Proof. The idea is the same as in the proof of Theorem 1, the only difference lying in the convolution identity at the start of the argument. More specifically, we seek an identity that expresses the triple convolution \(\Lambda_3\) as a linear combination of functions \(L^a \underbrace{\Lambda \ast \dots \ast \Lambda}_{b \text{ times}}\). We have \[\begin{align} \Lambda_3 &= L \Lambda_2 + \Lambda_2\ast \Lambda \\&=L (L \Lambda+ \Lambda\ast \Lambda) + (L \Lambda+ \Lambda\ast \Lambda) \ast \Lambda \\ &= L^2 \Lambda+ L (\Lambda\ast \Lambda ) +L \Lambda\ast \Lambda + \Lambda\ast \Lambda\ast \Lambda. \end{align}\] Noting that \(L (f \ast g )= ((Lf)\ast g)+ (f\ast (Lg))\) for any arithmetic functions \(f,g\), we infer that \(L(\Lambda \ast \Lambda)= 2 ( \Lambda \ast (L\Lambda))\), hence, \[\Lambda_3 - L^2 \Lambda -\frac{3}{2}L( \Lambda\ast \Lambda )= \Lambda\ast \Lambda\ast \Lambda.\] We can then deal with \(L^2 \Lambda\) by Lemma 10 and Theorem 2 with \(n=1=k_1\), while, \(L ( \Lambda\ast \Lambda )\) is dealt with by 14 and Lemma 10. We deduce that \[\sum_{\substack{ P\in \mathbb{Z}[t],|P|\leqslant H \\ \deg(P)=d}} \left| \sum_{\substack{1\leqslant m\leqslant x\\P(m)>0 } } (\Lambda \ast\Lambda \ast \Lambda)(P(m)) - \frac{1}{2} \mathfrak S_{ P}(x) x (\log H)^2 \right|^2 \ll \frac{H^{d+1} (x (\log H)^2)^2}{\log x}\] for \((\log H)(\log \log H) \leqslant x \leqslant(\log H)^\delta\). The terms with \(\omega(P(m))=1\) can be absorbed into the error term as was done in the proof of the proof of Theorem 1.
As \(\Lambda \ast\Lambda \ast \Lambda\) is supported on integers \(m\) with \(\omega(m)\leqslant 3\) it remains remains to show that the terms with \(\omega(P(m))=2\) contribute to the error term. We note that the triple convolution vanishes when evaluated at \(p^\alpha q^\beta\) for primes \(p\neq q\) and strictly positive integers \(\alpha,\beta\). Furthermore, the contribution of terms with \(\omega(P(m))=p^\alpha q^\beta\) and \(\alpha \geqslant 2,\beta\geqslant 2\) can be proved to be \(O(H^{d+1/2+\varepsilon})\) as in the end of the proof of Theorem 1.Since \((\Lambda \ast \Lambda \ast \Lambda)(p^2 q) =3 (\log p)^2\log q\), the contribution of the terms with \(P(m)=p^2 q\) is \[\ll \sum_{\substack{ P\in \mathbb{Z}[t],|P|\leqslant H \\ \deg(P)=d}} \left| \sum_{\substack{m\leqslant x\\P(m)\in \mathscr E} } (\Lambda \ast \Lambda \ast \Lambda)(P(m)) \right|^2 \ll x (\log H)^4 \sum_{\substack{ m \leqslant x \\ p \ll (x^dH)^{1/2} }} (\log p)^2 \sum_{\substack{ P\in \mathbb{Z}[t],|P|\leqslant H \\ \deg(P)=d,p^2 \mid P(m)\\ P(m)/p^2 \textrm{ prime}} } 1\] by the bounds \(\sum_{m\leqslant x} (\Lambda \ast \Lambda \ast \Lambda)(P(m)) \ll x (\log H)^3\) and \(\log (P(m)/p^2) \ll \log H\). In the sum over \(P\) in the right-hand side we write \(P=\sum_{j=0}^d c_j t^j\) and fix all \(c_j\) except \(c_0\). As \((x^d H)^{1/2}\ll H^{1/2+\varepsilon}\) for all \(\varepsilon\), the sum over \(c_0\) is trivially bounded by \(\ll 1+ H/p^2\), hence, the sum over all \(p> (\log H)^2\) gives the overall bound \[\ll x^2 (\log H)^4 \sum_{(\log H)^2 < p \ll (x^dH)^{1/2} }(\log p)^2 H^d (1+ H/p^2)\ll x^2 H^{d+1}(\log H)^4 ,\] which is satisfactory. In the remaining range \(p\leqslant(\log H)^2\) we fix all \(c_j\) except \(c_0\) and we bring in the variable \(N=\sum_{j\neq 0} c_j m^j\). Then the sum over \(c_0\) becomes \[\sharp\{|c_0| \leqslant H :p^2 \mid c_0+N, |c_0+N|/p^2 \textrm{ prime}\} = \sharp\left\{m\in \mathbb{Z}: \Big|m- \frac{N}{p^2}\Big| \leqslant\frac{H}{p^2}, |m| \textrm{ prime}\right\} \ll \frac{H}{p^2 \log H}\] by the Brun–Titchmarsh theorem. Thus, the overall contribution becomes \[\ll x^2 (\log H)^4 \sum_{ p \ll (x^dH)^{1/2} } (\log p)^2 \frac{H^{d+1} }{p^2 \log H} \ll x^2 H^{d+1} (\log H)^3.\] The first claim of Theorem 3 can now be proved by applying \[\omega(t)=3 \Rightarrow (\Lambda\ast \Lambda\ast \Lambda)(t)=6 \prod_{ \textrm{ prime } \ell \mid t} \log \ell =6 \mathscr L_2(t)\]with \(t=P(m)\). The second and third claim of Theorem 3 can be proved as in the analogous statements in Theorem 1. ◻
Define \(\Lambda^{(1)}=\Lambda\) and by induction for all positive integers \(a\) let \(\Lambda^{(a+1)}= \Lambda \ast \Lambda^{(a)}\). Our proofs of Theorem 1 and Theorem 3 start by making a passage to the \(\Lambda_k\) functions by an identity that expresses \(\Lambda_k\) as a finite linear combination of \(\Lambda^{(a)}L^b\) and then applying Theorem 2. In our last lemma, we show that this strategy does not work for detecting \(E_k\) numbers when \(k \geqslant 4\).
Lemma 11. For any integer \(k \geqslant 4\), the function \(\Lambda_k\) cannot be expressed as a finite linear combination of the functions \(\Lambda^{(a)}L^b\) for integers \(a,b\geqslant 0\) using real coefficients.
Proof. If \(\Lambda_k\) is a linear combination of \(L^a \Lambda^{(b)}\), then, by \(\Lambda_k=L \Lambda_{k-1} + \Lambda \ast \Lambda_{k-1}\) so is \(\Lambda \ast \Lambda_{k-1}\). Hence, there are \(c_{a,b}\in \mathbb{R}\) and \(N\in \mathbb{N}\) such that \[\Lambda \ast \Lambda_{k-1} = \sum_{0\leqslant a,b \leqslant N} c_{a,b} L^a \Lambda^{(b)}.\] Let \(p,q\) be two distinct primes and estimate both sides at \(pq\). We obtain that \[(\log p)(\log q)((\log p)^{k-2} + (\log q)^{k-2}) = (\Lambda \ast \Lambda_{k-1})(pq)\] equals \[\Lambda^{(2)} \sum_{0\leqslant a \leqslant N} c_{a,2} ( (\log p)+(\log q))^a =2 (\log p)(\log q) \sum_{0\leqslant a \leqslant N} c_{a,2} ( (\log p)+(\log q))^a\] because \(\Lambda^{(b)}(pq)=0\) when \(b\neq 2\). Letting \(x=\log p\) and \(y=\log q\) we infer that \[x^{k-2} + y^{k-2}= P(x+y)\] where \(P\) is a polynomial with coefficients in \(\mathbb{R}\).
If we fix any prime \(q\) then we fix \(y\) and the above identity can be seen as a polynomial identity in \(x\) that holds for infinitely many different real numbers, hence, it holds for all real numbers \(x\). The change of variables \(z=x+y\) leads to \[P(z)=(z-y)^{k-2}+ y^{k-2},\] which holds for all \(z\in \mathbb{R}\) and all \(y\) of the form \(\log q\), where \(a\) is a prime number. We can now get a contradiction as follows: If \(k\) is even then \(P(0)= 2 y^{k-2}\), thus, \(P(0)\) takes more than one value, a contradiction. If \(k\) is odd then \[\lim_{z\to 0} f'(z)=\lim_{z\to 0} (k-2)(z-y)^{k-3} =(k-2)(-y)^{k-3},\] which depends on \(y\) and therefore assumes more than a single value. ◻