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\def\RR{\mathbb{R}}
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\section*{Problem Set 7}

\begin{enumerate}[leftmargin=0.7cm]
\item Show that we cannot have $\liminf_{x\to \infty}\tfrac{\vf(x)}{x}<1$. (See Theorem 3 of Lecture 19 for context and hint.)
\item Prove that for $r>s$ non-negative integers, $\int_0^1\int_0^1\tfrac{x^r y^s}{1-xy}dx\,dy\in\tfrac{1}{d_r^2}\ZZ$ while $\int_0^1 \int_0^1\tfrac{x^r y^r}{1-xy}dx\,dy=\zeta(2)-\tfrac{1}{1^2}-\cdots -\tfrac{1}{r^2}$. [Hint: use the proof of Lemma 1 in Lecture 20.  This is easy.]
\item Prove $\zeta(2)$ (and hence $\pi$, since $\zeta(2)=\tfrac{\pi^2}{6}$) is irrational, by following the outline:

\noindent (a) Show that $\int_0^1\int_0^1\tfrac{(1-y)^n P_n(x)}{1-xy}dx\,dy=(-1)^n\int_0^1\int_0^1\tfrac{\phi(x,y)}{1-xy}dx\,dy$, where $\phi(x,y)=\tfrac{xy(1-x)(1-y)}{1-xy}$.

\noindent (b) Show that on $[0,1]^2$, $\phi$ is bounded by $(\tfrac{\sqrt{5}-1}{2})^5$.

\noindent (c) Using (2), parts (a) and (b), and Lemma 2 of Lecture 20, derive a contradiction from $\zeta(2)=\tfrac{P}{Q}$ ($P,Q\in \ZZ$, $Q\neq 0$).

\item Given a Dirichlet series $\sum_{n\geq 1}\tfrac{a_n}{n^s}$ (here $\{a_n\}\subset \CC$ is just some sequence) converging for some $s_0=\sigma_0+it_0$, it converges normally on the set $\{s\mid Re(s)>\sigma_0\}$.
\item Let $F(s)=\sum_{p}p^{-s}$, where the sum runs over all primes.  Show that $F(s)=\log\zeta(s)+G(s)$, where log denotes the principal branch of the logarithm, and $G(s)$ is analytic on the half-plane $Re(s)>\tfrac{1}{2}$. Deduce from this that the function $F(s)$ does \emph{not} have a meromorphic continuation to the left of the line $\sigma=1$.
\item Show that, if $x$ is sufficiently large, then the interval $[2,x]$ contains more primes than the interval $(x,2x]$.
\end{enumerate}

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