$
\definecolor{highIt}{RGB}{0, 180, 0}
\newcommand{\HighLight}[1]{\textcolor{highIt}{#1}}
$
$
\newcommand{\Complex}{{\mathbb C}}
\newcommand{\Real}{{\mathbb R}}
\newcommand{\Rat}{{\mathbb Q}}
\newcommand{\Nat}{{\mathbb N}}
\newcommand{\Int}{{\mathbb Z}}
\newcommand{\sss}{\scriptscriptstyle}
\newcommand{\dps}{\displaystyle}
\newcommand{\intll}[1]{{\mathfrak{f}_{#1}}}
\newcommand{\intul}[1]{{\mathfrak{g}_{#1}}}
\newcommand{\hb}[1]{{\widetilde{#1}}}
\newcommand{\bigc}{{\mathbb{A}}}
\newcommand{\smallc}{{c}}
\newcommand{\calO}{{\mathcal{O}}}
\newcommand{\bigO}[1]{{\calO\!\left( #1 \right)}}
\newcommand{\res}[1]{{\underset{#1}{\mathrm Res}}}
\newcommand{\resn}[2]{{ \left\{ \res{#1}{#2} \right\} }}
\newcommand{\de}[1]{\delta_{#1}}
\newcommand{\deb}[1]{\hb{\delta}_{#1}}
\newcommand{\der}[1]{\de{#1},\ldots,\de{k}}
\newcommand{\debr}[1]{\deb{#1},\ldots,\deb{k}}
\newcommand{\De}[1]{{\Delta_{#1}}}
\newcommand{\Dep}[1]{{\De{#1}\cdots\De{k}}}
\newcommand{\lambi}[1]{\lambda_{#1}}
\newcommand{\lambib}[1]{\hb{\lambda}_{#1}}
\newcommand{\lambip}{\lambi{\de{1}}\lambib{\deb{1}}}
\newcommand{\reals}[1]{{\sigma_{#1}}}
\newcommand{\realsb}[1]{{\hb{\sigma_{#1}}}}
%\newcommand{\lbs}{{{\scriptstyle \blacksquare}}}
\newcommand{\lbs}{{\varphi}}
\newcommand{\Eindi}[1]{{\gamma_{#1}}}
\newcommand{\sid}{{D}}
\newcommand{\sidb}{{\hb{D}}}
\newcommand{\tuple}[1]{{a_{#1}n\!+\!b_{#1}}}
\newcommand{\ktuple}{{\mathfrak{L}}}
\newcommand{\singse}{{\mathfrak{S}}}
\newcommand{\bombxqa}[4]{{\mathcal{E}_{#1}(#2;#3,#4)}}
\newcommand{\bombxq}[3]{{\mathcal{E}_{#1}^{\ast}\!\left(#2;#3\right)}}
\newcommand{\bombA}{{\mathsf{A}}}
\newcommand{\bombB}{{\mathsf{B}}}
\newcommand{\bombq}[1]{{\frac{\sqrt{#1}}{\log^\bombB(#1)}}}
\newcommand{\bombqa}[2]{{\frac{\sqrt{#1}}{{#2}\log^\bombB(#1)}}}
\newcommand{\sumbomb}[1]{{\sum_{q\leq\bombq{#1}}}}
\newcommand{\logf}[2]{{\log\!\dfrac{#1}{#2}}}
\newcommand{\logfd}[3]{{\left(\logf{#1}{#2}\right)^{\scriptscriptstyle #3}}}
\newcommand{\logat}{{\log^{\bigc}\!T}}
\newcommand{\logNd}[1]{{(\log\!N)^{#1}}}
\newcommand{\logzd}[1]{{(\log\!z)^{#1}}}
\newcommand{\logdN}[1]{{\log^{#1}\!N}}
\newcommand{\logdz}[1]{{\log^{#1}\!z}}
\newcommand{\ze}[2]{{\zeta^{#2}({#1})}}
\newcommand{\pfact}[4]{{\left( 1{#1}\frac{#2}{p^{#3}}\right)^{#4}}}
\newcommand{\conmo}[3]{{{#1}\equiv{#2}\pmod{#3}}}
\newcommand{\pprod}[1]{{p_1\!\cdots\!p_{#1}}}
\newcommand{\aNover}[1]{{\frac{a_1N}{#1}}}
\newcommand{\aaNover}[1]{{\frac{2a_1N}{#1}}}
\newcommand{\zfp}[2]{{z^{\frac{#1}{#2}}}}
\newcommand{\Nfp}[2]{{N^{\frac{#1}{#2}}}}
\newcommand{\sitrun}[1]{z^{#1/\lbs}}
\newcommand{\dote}{{\doteqdot}}
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\newcommand{\varR}{{\mathcal{R}}}
\newcommand{\flip}{{[\sid\rightleftarrows\sidb]}}
\newcommand{\vect}[2]{{\mathbf{#1_{#2}}}}
\newcommand{\pvect}[1]{{\vect{p}{#1}}}
\newcommand{\yvect}[1]{{\vect{y}{#1}}}
\newcommand{\pntR}[1]{{\mathfrak{R}(#1)}}
\newcommand{\psum}[1]{{\sideset{}{'}\sum_{f_{#1}(\pvect{#1})}^{g_{#1}(\pvect{#1})}}}
\newcommand{\halfp}[1]{{\mathbb{H}_{#1}}}
\newcommand{\result}[3]{{\lceil{#1}:{#2}\,,{#3}\rfloor}}
\newcommand{\cinf}{{\mathcal{C}^{\infty}}}
\newcommand{\ncr}[2]{{\mbox{\large$\mathrm{C}$}_{\sss #2}^{\sss #1}}}
\newcommand{\sumii}[2]{{\sum_{\begin{subarray}{c}
{#1}\\
{#2}
\end{subarray}}}}
\newcommand{\sumiii}[3]{{\sum_{\begin{subarray}{c}
{#1}\\
{#2}\\
{#3}
\end{subarray}}}}
\newcommand{\sumiv}[4]{{\sum_{\begin{subarray}{c}
{#1}\\
{#2}\\
{#3}\\
{#4}
\end{subarray}}}}
\newcommand{\sumv}[5]{{\sum_{\begin{subarray}{c}
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{#2}\\
{#3}\\
{#4}\\
{#5}
\end{subarray}}}}
\newcommand{\prodi}[1]{\prod_{#1}}
$
-
In the thesis, the parameter $\lbs$ determines how big the support of the sieve is.
The bigger the support, the better would be the results.
The allowed values for $\lbs$ are related to our knowledge of the "level of distribution".
-
On page 27, in the statement of Lemma 2.4.3, the imaginary parts of all of the domains
should be truncated to $[-T_0, T_0]$, where $T_0 >0$ is an unspecified constant. For example,
I should change
$$
\prodi{i=1}^{2\kappa}\halfp{-a}
$$
to
$$
\{\sigma+it: t\in[-T_0, T_0]\} \cap \left(\prodi{i=1}^{2\kappa}\halfp{-a}\right),
$$
and do the same thing for other domains in the statement.
It is because in the proofs of Lemma 3.1.1 and Lemma 4.1.1, in which this lemma is applied,
I need to ensure that the zeta function is in a zero-free region so that
$$
\dfrac{1}{s\zeta(1+\lbs s)}
$$
on the domain of consideration is a $\cinf$-function. The truncation corresponds to this requirement.
-
On page 44, in the proof of Lemma 2.5.5(b), it should be
$$
\begin{eqnarray*}
& & \sumbomb{x}\max_{(a,q)=1}
\left|
\sumiii{x<m\leq2x}{\conmo{m}{a}{q}}{\scriptsize{\mbox{$m$ not square free}}}h(m)
-\frac{1}{\phi(q)}\sumiii{x<m\leq2x}{(m,q)=1}{\scriptsize{\mbox{$m$ not square free}}}h(m)
\right|\\
&\leq & \sumbomb{x}\max_{(a,q)=1}
\sum_{y<n\leq\HighLight{\sqrt{2x}}}\sumii{\frac{x}{n^2}<\HighLight{v}\leq\frac{2x}{n^2}}{\conmo{\HighLight{v}n^2}{a}{q}}\hspace{-0.3cm}1
+
\sumbomb{x}
\frac{1}{\phi(q)}\sum_{y<n\leq\HighLight{\sqrt{2x}}}\sum_{\frac{x}{n^2}<\HighLight{v}\leq\frac{2x}{n^2}}\hspace{-0.2cm}1
\end{eqnarray*}
$$
because $m$ can be a square of a prime and $n$ is treated as the least prime factor of $m$ such that $n^2|m$.
The same result still follows, as the above is
$$
\begin{eqnarray*}
&\ll &
\sumbomb{x}
\sum_{y<n\leq\sqrt{2x}}\left(\frac{x}{n^2q}+1\right)
+
\sumbomb{x}
\frac{1}{\phi(q)}\sum_{y<n\leq\infty}\frac{x}{n^2}\\
&\ll &
\frac{x\log\!x}{y}
+ \frac{x}{\log^\bombB\!x}
+ \frac{x\log\!^2x}{y}.
\end{eqnarray*}
$$
This is $\ll \dfrac{x}{\log^\bombA\!x}$ for suitable $\bombB$.
-
On page 48, 2nd last line, I have used the following estimate:
$$
\left| \frac{1}{\ze{1+\lbs s_i}{}} \right|
=
\left| \sum\frac{\mu(n)}{n^{1+\lbs s_i}} \right|
\leq \sum\frac{1}{n^{1+\lbs \reals{i}}}
= \ze{1+\lbs \reals{i}}{}.
$$
-
In the proofs of Lemma 3.1.1 and Lemma 4.1.1, I have used the following estimate: there exists an absolute constant $C_0$ such that
$$
\left| \frac{1}{s\ze{1+\lbs s}{}} \right|
\ll\left\{ \begin{array}{rcl}
1 & \mbox{for}
& |t|\leq C_0 \\
\frac{\log|t|}{|t|} & \mbox{for} & |t|>C_0.
\end{array}\right.
$$
For small $|t|$ it is because the zeta function is in a zero-free region, while for large $|t|$ it is because of Lemma 2.3.3.
-
On page 61, in the statement of Theorem 3.3.2, the implied constant
should depend on $\eta$ too. It is because when I use Lemma 3.3.1 in the proof,
I have used the estimation
$$
\bigO{ \dfrac{1}{Y(\log N)} } \ll_\eta \dfrac{1}{(\log N)^{1.5}}.
$$
This mistake should be immaterial to the main results of the thesis.
-
In the first equation on page 65, the following highlighted conditions are dropped:
$$
\begin{eqnarray*}
Q_2 \; & \dote & \sumii{a_1N<\ell\leq2a_1N}{\HighLight{(\ell, A)=1}}
\frac{\Eindi{2}(\ell)}{\phi(a_1)}
\sumii{\de{2},\deb{2}}{(\De{2},A\ell)=1}
\dfrac{[\lambi{1}\lambib{1}]}
{\phi(\De{2})}\\
& & + \sum_{Y\!<p_1\leq\sitrun{1}}
\sumii{\aNover{p_1}<\ell\leq\aaNover{p_1}}{\HighLight{(p_1\ell, A)=1}}
\frac{\varpi(\ell)}{\phi(a_1)}
\sumii{\de{2},\deb{2}}{(\De{2},Ap_1\ell)=1}
\dfrac{[\lambi{1}\lambib{p_1}+\lambi{p_1}\lambib{1}+\lambi{p_1}\lambib{p_1}]}
{\phi(\De{2})}.
\end{eqnarray*}
$$
It is because every prime factor of $\pprod{r}\,\ell$ is greater than $Y$, so the conditions are satisfied automatically.
-
On page 70, I say that the number of non-zero $\lambi{}$ is about $N^{0.4}$, which may be confusing.
I have been conceptualizing $\lambi{}$ as a multiset,
so what I refer to is the number of elements of the multiset
$$
\left\{
\lambi{\de{1},\de{2}}^{\mathbf{\sid}}: \;\;
\de{1}, \de{2}\in\Nat, \;\;
\lambi{\de{1},\de{2}}^{\mathbf{\sid}} \neq 0
\right\}.
$$
Using the set notations, this number is just
$$
\# \left\{
(\de{1}, \de{2})\in \Nat^2: \;\;
\lambi{\de{1},\de{2}}^{\mathbf{\sid}} \neq 0
\right\} =: C_0.
$$
The size $N^{0.4}$ is due to the following estimates. On the one hand,
$$
C_0 \leq z^{1/5}\cdot z^{1/5} \leq N^{1/5}\cdot N^{1/5} = N^{0.4}.
$$
On the other hand, as square-free integers have density $6/\pi^2$, we have
$$
C_0 \gg z^{1/5}\cdot z^{1/5} \gg N^{0.4-\varepsilon}.
$$
-
On page 100, 3rd-to-last line, instead of
$$
f_0(\yvect{i})<y_i\leq g_i(\yvect{i}),
$$
it should be
$$
f_{i-1}(\yvect{i-1})<y_i\leq g_{i-1}(\yvect{i-1}).
$$
The same mistake occurs on page 106, line 6. It should not affect the proofs.