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\textbf{18.785 Number Theory\hspace{228pt}Fall~2017}\\\vspace{4pt}
\textbf{Problem Set \#8\hspace{220pt}Due: 11/13/2017}
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% add k-divisor function 1^{*k} to problem 4

\subsection*{Description}

These problems are related to the material covered in Lectures 16--18.
Your solutions are to be written up in latex (you can use the latex source for the problem set as a template) and submitted as a pdf-file with a filename of the form \texttt{SurnamePset8.pdf} via e-mail to \texttt{drew@math.mit.edu} by {\bf noon} on the date due.
Collaboration is permitted/encouraged, but you must identify your collaborators, and any references you consulted.
If there are none, write ``\textbf{Sources consulted:\ none}" at the top of your problem set.
The first person to spot each non-trivial typo/error in any of the problem sets or lecture notes will receive 1--5 points of extra credit.
\medskip

\noindent
\textbf{Note:} Problem 1 uses complex analysis beyond the quick review given in the notes.
\medskip

\noindent
\textbf{Instructions:} First do the warm up problems, then pick two of problems 1--5 to solve and write up your answers in latex.
Finally, complete the survey problem 6.

\subsection*{Problem 0.}
These are warm up problems that do not need to be turned in.
\begin{enumerate}
\setlength\itemsep{0pt}
\item[(a)] In class we gave an elementary proof that $\vartheta(x)=O(x)$.  Give a similarly elementary proof that $x=O(\vartheta(x))$ (both bounds were proved by Chebyshev before the PNT).
\item[(b)] Prove the M\"obius inversion formula, which states that if $f$ and $g$ are functions $\Z_{\ge 1}\to \C$ that satisfy $g(n)=\sum_{d|n}f(d)$ then $f(n)=\sum_{d|n}\mu(d)g(n/d)$, where $\mu(n):=(-1)^{\#\{p|n\}}$ if $n$ is squarefree and $\mu(n)=0$ otherwise.
\item[(c)] Verify that for all Schwartz functions $f,g\in \mathcal{S}(\R)$ we have
\[
\widehat{f*g}=\hat f \hat g,\qquad\text{and}\qquad\widehat{fg}=\hat f*\hat g.
\]
(the Fourier transform turns convolutions into products and vice versa).
\end{enumerate}

\subsection*{Problem 1. The explicit formula (49 points)}

Let $Z(s):=\pi^{-s/2}\Gamma(\frac{s}{2})\zeta(s)$ denote the completed zeta function; we proved in class that it has the integral representation
\[
Z(s)=\int_1^\infty\sum_{n=1}^\infty e^{-\pi n^2x}(x^{s/2}+x^{(1-s)/2})\frac{dx}{x}-\frac{1}{s}-\frac{1}{1-s},
\]
and extends to a meromorphic function on $\C$ with functional equation $Z(s)=Z(1-s)$.

Recall Hadamard's Factorization Theorem: if $f(s)$ is an entire function and $n$ is an integer for which there exists a positive $c < n+1$ such that $|f(s)|=O\bigl(\exp(|s|^c))$ then
\begin{equation}\label{eq:hft}
f(s) = s^m e^{g(s)}\prod_\rho\left(1-\frac{s}{\rho}\right)E_n\left(\frac{s}{\rho}\right),
\end{equation}
where $m=\ord_0(f)$,  $g\in \C[s]$ has degree at most $n$, the product ranges overs zeros $\rho\ne 0$ of $f(s)$ (with multiplicity), and $E_n(z)=\exp(\sum_{k=1}^n\frac{z^k}{k})$.

\begin{enumerate}[{\bf(a)}]
\item Prove that we can apply \eqref{eq:hft} to $f(s):=s(s-1)Z(s)$ with $n=1$ and $m=0$.
\item  Prove that we can apply \eqref{eq:hft} to $f(s):=\Gamma(s)^{-1}$ with $n=1$ and $m=1$.
\item Prove that
\[
(s-1)\zeta(s) =e^{a+bs}\prod_\rho\left(1-\frac{s}{\rho}\right)e^{\frac{s}{\rho}}\prod_{n=1}^\infty\left(1+\frac{s}{2n}\right)e^{-\frac{s}{2n}}
\]
for some $a,b\in \C$, where $\rho$ ranges over the zeros of $\zeta(s)$ in the critical strip.
\item Using (c) and the Euler product for $\zeta(s)$, show that $b=\frac{\zeta'(0)}{\zeta(0)}-1$ and
\[
\sum_p\sum_{m\ge 1}p^{-ms}\log p = \frac{s}{s-1}-\frac{\zeta'(0)}{\zeta(0)}-\sum_\rho\left(\frac{1}{s-\rho}+\frac{1}{\rho}\right) -\sum_{n\ge 1}\left(\frac{1}{s+2n}-\frac{1}{2n}\right)
\]
on $\Re(s)>1$.
\end{enumerate}

\noindent
We now recall the identity
\[
\frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}\frac{x^s}{s}ds = \begin{cases}
1 & \text{if }x>1,\\
0 & \text{if }0<x<1,
\end{cases}
\]
valid for any $x,\sigma>0$, and define the \emph{Perron integral transform}
\[
f\mapsto \lim_{t\to\infty}\frac{1}{2\pi i}\int_{\sigma-it}^{\sigma+it}f(s)\cdot\frac{x^s}{s}ds.
\]
We also define an alternative version of Chebyshev's function
\[
\psi(x):=\sum_{p^n\le x}\log p,
\]
where the sum is over all prime powers $p^n\le x$ (but note that we take $\log p$ not $\log p^n$).

\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{4}
\item Fix $x>1$ not a prime power.
By applying the Perron integral transform to both sides of the equation in (d), and assuming that the RHS can be computed by applying Cauchy's residue formula term by term to the sums (and that the Perron integral transform converges in each case), deduce the \emph{explicit formula}
\[
\psi(x) = x-\sum_\rho \frac{x^\rho}{\rho}-\frac{\zeta'(0)}{\zeta(0)} +\sum_n\frac{x^{-2n}}{2n},
\]
where $\rho$ ranges over the the non-trivial zeros of $\zeta(s)$.
\item Fix $c\in [1/2,1)$ and suppose that $\zeta(s)$ has no zeros in the strip $c<\Re(s)<1$. 
Assume that the number of zeros $\rho$ with $|\Im(\rho)|\le T$ is bounded by $O(T\log T)$.
Derive the following bounds: $\psi(x)=x+ O(x^{c+\epsilon})$, $\vartheta(x)=x+O(x^{c+\epsilon})$, and $\pi(x)=\Li(x)+O(x^{c+\epsilon})$, for any $\epsilon > 0$ (in fact one replace $\epsilon$ with $o(1)$).
\end{enumerate}

\begin{remark}
The explicit formula you obtained in (e) is a slight variation of the one given by Riemann (who also glossed over the somewhat delicate convergence issues you were told to ignore ---  to make this rigorous you actually need to specify the order in which the sum over $\rho$ is computed, it does not converge absolutely).  It is worth noting that even with the explicit formula in hand,  Riemann was unable to prove the prime number theorem because he could not (and we still cannot) prove that one can take $c<1$ in (f).
\end{remark}

\subsection*{Problem 2. The zeta function of $\Fq[t]$ (49 points)}

Recall that for a number field $K$, the Dedekind zeta function $\zeta_K(s)$ is defined by
\[
\zeta_K(s):=\sum_I \N(I)^{-s},
\]
where $I$ ranges over nonzero ideals of $\O_K$ and $\N(I)$ is the absolute norm, which is just the cardinality of the residue field $\O_K/\p$ when $I$ is a prime ideal $\p$.

The definition of $\N(\p):=\#\O_K/\p$ as the cardinality of the residue field makes sense in any global field and extends  multiplicatively to all $\O_K$-ideals.
In this problem you will investigate the zeta function $\zeta_q(s):=\sum_I \N(I)^{-s}$, where $I$ ranges over ideals of the ring of integers $\O_K:=\Fq[t]$ of the rational function field $K=\Fq(t)$.

\begin{remark}
The zeta function $\zeta_K$ of a global function field $K$ defined in Problem 5 differs from the zeta function of its ring of integers $\O_K$, which is what we are considering here.
\end{remark}

\begin{enumerate}[{\bf(a)}]
\item Show that every $\O_K$-ideal has the form $I=(f)$, with $f\in \Fq[x]$ monic, and then $\N(I)=\#(\O_K/f\O_K)=q^{\deg f}$.
Then prove that $\zeta_q(s)=\frac{1}{1-q^{1-s}}$ for $\Re(s)>1$.
\item Prove that $\zeta_q(s)$ has the Euler product
\[
\zeta_q(s)=\prod_\p(1-\N(\p)^{-s})^{-1},
\]
valid for $\Re(s)>1$.
\item Prove that $\zeta_q(s)$ extends to a meromorphic function on $\C$ with a simple pole at $s=1$ and no zeros.  Give the residue of the pole at $s=1$.
\item Define a completed zeta function $Z(s)=G(s)\zeta_q(s)$, where $G(s)$ is some suitably chosen meromorphic function, and prove that your completed zeta function satisfies the functional equation
\[
Z(s)=Z(1-s).
\]
\item Let $a_d$ denote the number of irreducible monic polynomials in $\Fq[x]$ of degree $d$.  Using (b) and~(c), prove that 
\[
\sum_{d|n}da_d=q^n.
\]
and use this to derive an explicit formula for $a_n$.
\item Prove the prime number theorem for $\Fq[t]$, which states that
\[
a_n=\frac{q^n}{n}+O\left(\frac{1}{n}q^{n/2}\right).
\]
\end{enumerate}
\begin{remark}
The error term in (f) is comparable to the error term in the PNT under the Riemann hypothesis (replace $q^n$ with $x$); note that the analog of the Riemann hypothesis for $\zeta_q(s)$ is (vacuously) true, by (c).
\end{remark}
\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{6}
\item Let $S(n)$ be the set of monic polynomials of degree $n$ in $\Fq[t]$, and let $I(n)$ be the subset of polynomials in $S(n)$ that are irreducible. Show that $\#I(n)/\#S(n)\sim \frac{1}{n}$.
Now let $R(n)$ be the subset of polynomials in $S(n)$ that have no roots in $\Fq$.  Give an asymptotic estimate for $\#R(n)/\#S(n)$.
\item Let $Q(n)$ denote the subset of $S(n)$ consisting of squarefree polynomials.  Prove $\lim_{n\to\infty}\#Q(n)/\#S(n)=1/\zeta_q(2)$ and derive an asymptotic estimate for this limit.
\item For nonzero $f\in \O_K$ define $\Phi$ via $\Phi(f):=\#(\O_K/f\O_K)^\times$.  Prove the following
\begin{enumerate}[1.]
\item $\Phi(f)=\N(f)\prod_{p|f}(1-\N(p)^{-1})$, where $p$ ranges over the irreducible factors of $f$.
\item For all $f,g\in \O_K$ with $(f,g)=1$ we have $g^{\Phi(f)}\equiv 1\bmod f$.
\end{enumerate}
\end{enumerate}

\subsection*{Problem 3. Bernoulli numbers (49 points)}

For integers $n\ge 0$, the \emph{Bernoulli polynomials} $B_n(x)\in \Q[x]$ are defined as the coefficients of the exponential generating function
\[
E(t,x):=\frac{te^{tx}}{e^t-1} = \sum_{n\ge 0}\frac{B_n(x)}{n!}t^n.
\]
The \emph{Bernoulli numbers} $B_n\in \Q$ are defined by $B_n=B_n(0)$.

\begin{enumerate}[{\bf(a)}]
\item Prove that $B_0(x)=1$, $B_n'(x)=nB_{n-1}(x)$, and $B_n(1)=B_n(0)$ for $n\ne 1$, and that these properties uniquely determine the Bernoulli polynomials.
\item Prove that $B_n(x+1)-B_n(x)=nx^{n-1}$ and
\[
B_n(x+y)=\sum_{k=0}^n\binom{n}{k}B_k(x)y^{n-k}.
\]
Use this to show that $B_k$ can alternatively be defined by the recurrence $B_0=1$ and
\[
B_n = -\frac{1}{n+1}\sum_{k=0}^{n-1}\binom{n+1}{k}B_k
\]
for all $n>0$, and show that $B_n=0$ for all odd $n>1$.
\item Recall the hyperbolic cotangent function $\coth z := \frac{e^z+e^{-z}}{e^z-e^{-z}}$.
Prove that
\[
z\coth z = \sum_{n\ge 0} B_{2n}\frac{(2z)^{2n}}{(2n)!}.
\]
\item Show that $\cot z = i \coth iz$ and then derive (as Euler did) the identity
\[
z\cot z = 1-2\sum_{k\ge1}\frac{z^2}{k^2\pi^2-z^2}.
\]
\item Use (c) and (d) to prove that for all $n\ge 1$ we have
\[
\zeta(2n) = (-1)^{n-1}\frac{(2\pi)^{2n}B_{2n}}{2\cdot(2n)!},
\]
and then use the functional equation to prove that for all $n\ge 1$ we have
\[
\zeta(-n) = - \frac{B_{n+1}}{n+1}.
\]
\item Prove (rigorously!) that for any integer $n>1$ the asymptotic density of integers that are $n$-\emph{power free} (not divisible by $p^n$ for any prime $p$) is $1/\zeta(n)$ and compute this density explicitly for $n=2,4,6$.
\item Prove that for all integer $n,N > 1$ we have
\[
\sum_{m=0}^{N-1}(m+x)^{n-1} = \frac{B_n(N+x)-B_n(x)}{n}.
\]
Use this to deduce Faulhaber's formula
\[
P_n(N):=\sum_{m=1}^{N-1} m^n = \frac{1}{n+1}\sum_{k=0}^n\binom{n+1}{k}B_kN^{n+1-k}
\]
for summing $n$th powers.  Compute the polynomials $P_n(N)$ explicitly for $n=2,3,4$.
\end{enumerate}

\subsection*{Problem 4. Arithmetic functions and Dirichlet series (49 points)}

Recall that an \emph{arithmetic function} is a function $a\colon \Z_{\ge 1}\to \C$; we say that $a\ne 0$ is \emph{multiplicative} if $a(mn)=a(m)a(n)$ holds for all relatively prime $m,n$, and \emph{totally multiplicative} if this holds for all $m,n$.
Below are some examples; as usual,~$p$ denotes a prime, $p^e$ denotes a (nontrivial) prime power, and $d|n$ indicates that~$d$ is a positive divisor of $n$.
\begin{itemize}
\setlength{\itemsep}{0pt}
\item $0(n)=0$,\ \ $1(n)=1$,\ \ $\id(n):=n$, \ \ $e(n):=0^{n-1}$;
\item $\tau(n):=\#\{d|n\}$, \ \ $\sigma(n):=\sum_{d|n}d$;
\item $\omega(n):=\#\{p|n\}$,\ \  $\Omega(n):=\#\{p^e|n\}$, \  $\phi(n):=\#(\Z/n\Z)^\times$;
\item $\lambda(n):=(-1)^{\Omega(n)}$,\ \ $\mu(n):=(-1)^{\omega(n)}\cdot 0^{\Omega(n)-\omega(n)}$, \ \  $\mu^2(n):=\mu(n)^2$.
\end{itemize}
The set of all arithmetic functions forms a $\C$-vector space that we denote $\A$.
Associated to each arithmetic function is a \emph{Dirichlet series} $\sum_{n\ge 1}c_nn^{-s}$ defined by
\[
D_a(s):=\sum_{n\ge 1}a(n)n^{-s}
\]
The \emph{Dirichlet convolution} $a*b$ of arithmetic functions $a$ and $b$ is defined by
\[
(a*b)(n) := \sum_{d|n}a(d)b(n/d),
\]
We use $f^{*n}$ to denote the $n$-fold convolution $f*\cdots *f$.

\begin{enumerate}[{\bf(a)}]
\item Prove that for any arithmetic functions we have $D_{a*b}=D_aD_b$ and that endowing~$\A$ with a multiplication defined by Dirichlet convolution makes $\A$ a $\C$-algebra that is isomorphic to the $\C$-algebra of Dirichlet series (with the usual multiplication).
\item Show that $\A$ is a local ring with unit group $\A^\times=\{f\in \A:f(1)\ne 0\}$ and maximal ideal $\A_0=\{f\in \mathcal A:f(1)=0\}$.  Prove that the set of multiplicative functions $\mathcal M$ forms a subgroup of $\mathcal A_1:=\{f\in \A:f(1)=1\}\subseteq \A^\times$. Is this also true of the set of totally multiplicative functions?
\item Prove the following identities $\mu*1=e$, $\phi*1=\id$, $\mu*\id=\phi$, $1*1=\tau$, $\id*1=\sigma$.
Use $\mu*1=e$ to give a one-line proof of the M\"obius inversion formula.
\item  Define the exponential map $\exp\colon\A\to \A$ by
\[
\exp(f):=\sum_{n=0}^\infty \frac{f^{*n}}{n!} = e + f + \frac{f*f}{2} + \cdots
\]
Prove that $\exp$ defines a group isomorphism from $(\A_0,+)$ to $(\A_1,*)$ with inverse
\[
\log(f):=\sum_{n=1}^\infty \frac{(-1)^{n-1}(f-e)^{*n}}{n},
\]
\item Define $\kappa(n)$ to be $1/k$ if $n=p^k$ is a prime power and $0$ otherwise.
Prove that $\exp\kappa=1$, and deduce that $\exp(-\kappa)=\mu$ and $\exp(2\kappa)=\tau$.
\item Prove that each $f\in \A_1$ has a unique square-root $g\in \A_1$ for which $g^{*2}=f$ that we denote $f^{*1/2}$.
Prove that $1^{*1/2}=\exp(\kappa/2)$ and compute $\exp(\kappa/2)(n)$ for $n$ up to $10$.
\end{enumerate}

\subsection*{Problem 5. The Weil conjectures for global function fields (49 points)}

Let $K/\Fq(t)$ be a global function field, with $\Fq$ algebraically closed in $K$.
The \emph{divisor group} $\Div K$ is the free abelian group generated by the places of $K$; it consists of formal sums $\sum_P n_PP$ over $P\in M_K$ in which only finitely many $n_P\in\Z$ are nonzero.
This is the same as the group of $M_K$-divisors we defined in Lecture 15, but here we view $\Div K$ as an additive group and use $P$ to denote a place rather than $v$.

Corresponding to each $f\in K^\times$ we have a \emph{principal divisor}
\[
\div(f) := \sum_P \ord_P(f)P,
\]
where $\ord_P\colon K^\times\to \Z$ is the discrete valuation corresponding to $P$, which we extend to $\Div K$ by defining $\ord_P(\sum_P n_P P)=n_P$.  Two divisors $D_1$ and $D_2$ are \emph{linearly equivalent} if $D_1-D_2$ is a principal divisor, and we write $D_1\sim D_2$ to indicate this; this defines an equivalence relation on $\Div K$ and we use $[D]$ to denote the equivalence class of $D$.

The \emph{degree} of a place $P$ is the dimension of the residue field of the local field $K_P$ as an $\Fq$-vector space; it extends to a group homomorphism $\deg\colon \Div K\to \Z$ whose kernel contains the subgroup of principal divisors (by the product formula); the corresponding quotient is denoted $\Pic^0 K$.
The \emph{norm} of a divisor $D$ is defined by $\N(D):=q^{\deg D}$; when $D=P$ this is the cardinality of the residue field and if $D$ is supported only on finite places this agrees with the absolute norm defined in Problem 2.

We partially order divisors by defining
\[
D_1\le D_2 \qquad\Longleftrightarrow\qquad \ord_P(D_1)\le \ord_P(D_2)\text{ for all } P\in M_K.
\]
A divisor $D\ge 0$ is said to be \emph{effective}.
The \emph{zeta function} of $K$ is defined as a sum over effective divisors
\[
\zeta_K(s):=\sum_{D\ge 0}\N(D)^{-s}=\sum_{D\ge 0}q^{-s\deg(D)}
\]
The \emph{Weil conjectures} (for global function fields) concern three properties of $\zeta_K(s)$:
\begin{itemize}
\item $\zeta_K(s)$ is a rational function of $q^{-s}$.
\item There is a functional equation that relates $\zeta_K(1-s)$ and $\zeta_K(s)$.
\item The zeros of $\zeta_K(s)$ all lie on the line $\Re(s)=1/2$.
\end{itemize}
In this problem you will prove the first two; Weil proved the third in the 1940s, and a generalization to algebraic varieties of higher dimension conjectured by Weil was proved by Deligne in the 1970s.
\smallskip

Associated to each divisor $D\in \Div K$ is a \emph{Riemann-Roch space}
\[
L(D) := \{f\in K^\times:\div(f)\ge -D\} \cup \{0\},
\]
which is an $\Fq$-vector space whose finite dimension we denote $\ell(D)\in\Z_{\ge 0}$.
The degree $\deg D$ and dimension $\ell(D)$ of a divisor $D$ depend only on the divisor class $[D]$ and are related by the following theorem (which you are not asked to prove).

\begin{theorem}[Riemann-Roch]
Let $K$ be a global function field.
There is an integer $g\ge \Z_{\ge 0}$ and divisor $C\in \Div K$ such that for all divisors $D\in \Div K$ we have
\[
\ell(D)=\deg(D)-g+1+\ell(C-D).
\]
\end{theorem}

\begin{enumerate}[{\bf(a)}]
\item Prove that $\ell(C)=g$ and $\deg(C)=2g-2$, and that for $\deg(D)\ge 2g-2$ we have $\ell(D)=\deg(D)-g+1$ unless $D\sim C$.  Conclude that both the integer $g$ (the \emph{genus}) and the divisor class $[C]$ (the \emph{canonical class}) are uniquely determined.
\item Prove that for any $n\ge 0$ the number of effective divisors of degree $n$ is finite, and the number of divisor classes of degree $n$ is finite (so in particular, the group $\Pic^0 K$ of divisor classes of degree $0$, is finite).
\item Prove that for any divisor $D$ the number of effective divisors in $[D]$ is $\frac{q^{\ell(D)}-1}{q-1}$.
\item Prove that sum defining $\zeta_K(s)$ converges on $\Re(s)>1$ and we have an Euler product
\[
\zeta_K(s)=\prod_P(1-\N(P)^{-s})^{-1}.
\]
\item Let $a_n$ be the number of effective divisors of degree $n$.  Prove that
\[
a_n=\sum_{\deg([D])=n}\frac{q^{\ell(D)}-1}{q-1},
\]
where the sum is over the divisor classes of degree $n$, and show that if we define $Z_k(u):=\sum_{n\ge 0}a_nu^n$ then $\zeta_K(s)=Z_K(q^{-s})$ for $\Re s>1$.
\item Prove that there is a polynomial $L_K\in\Z[u]$ of degree $2g$ for which
\[
Z_K(u)=\frac{L_K(u)}{(1-u)(1-qu)},
\]
and it satisfies $L_K(0)=1$ and $L_K(1)=\#\Pic^0K$.
\item Prove that $Z_K(q^{-s})$ is meromorphic on $\C$ and thus provides an analytic continuation of $\zeta_K(s)$ to $\C$ with simple poles at $s=0,1$.  Are these the only poles?
\item Let $\xi_K(s):=q^{(g-1)s}\zeta_K(s)$.  Prove that $\xi_K(s)$ satisfies the functional equation
\[
\xi_K(1-s)=\xi_K(s).
\]
\end{enumerate}

\subsection*{Problem 6. Survey (2 points)}
Complete the following survey by rating each problem you attempted on a scale of 1 to~10 according to how interesting you found it (1 = ``mind-numbing," 10 = ``mind-blowing"), and how difficult you found it (1 = ``trivial," 10 = ``brutal").  Also estimate the amount of time you spent on each problem to the nearest half hour.

\begin{center}
\begin{tabular}{l|r|r|r|}
& Interest & Difficulty & Time Spent\\\hline
Problem 1 & & & \\\hline
Problem 2 & & & \\\hline
Problem 3 & & & \\\hline
Problem 4 & & & \\\hline
Problem 5 & & & \\\hline
\end{tabular}
\end{center}
\noindent
Please rate each of the following lectures that you attended, according to the quality of the material (1=``useless", 10=``fascinating"), the quality of the presentation (1=``epic fail", 10=``perfection"), the pace (1=``way too slow", 10=``way too fast", 5=``just right") and the novelty of the material to you (1=``old hat", 10=``all new").

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\begin{tabular}{l|l|r|r|r|r|r}
Date & Lecture Topic & Material & Presentation & Pace & Novelty\\\hline
11/8 & The functional equation & & & & \\\hline
11/10 & Dirichlet L-functions & & & & \\\hline
\end{tabular}
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\noindent
Please feel free to record any additional comments you have on the problem sets and the lectures, in particular, ways in which they might be improved.


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