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\section*{Problem Set 6}

\begin{enumerate}[leftmargin=0.7cm]
\item Determine $\Gamma(\tfrac{1}{6})$ in terms of $\Gamma(\tfrac{1}{3})$.

\item Define the \emph{digamma} function $\Psi(z):=\Gamma'(z)/\Gamma(z)$. 

\noindent (a) Check that $\Psi(1+z)=\Psi(z)+\tfrac{1}{z}$.

\noindent (b) The Weierstrass product formula for $\Gamma(z)$ yields a summation formula for $\log \Gamma(z)$. By differentiating this formula, and using $\tfrac{1}{1+x}-1=-\sum_{k\geq 2}(-1)^k x^{k-1}$, derive the power series expansion $\Psi(1+z)=-\gamma+\sum_{k\geq 2}(-1)^k \zeta(k)z^{k-1}$.

\noindent (c) Use this to obtain a power series for $\log \Gamma(1+z)$. By evaluating this power series at $z=1$, obtain a formula for the Euler constant $\gamma$ in terms of the zeta values $\zeta(k)$.

\item (a) Compute $\Gamma(\tfrac{1}{2}-n)$ for $n \in \mathbb{Z}_{>0}$.

\noindent (b) Show that $\lim_{n\to \infty}\Gamma(\tfrac{1}{2}-n+it)= 0$ uniformly for $t\in \RR$.

\item Show that for $x>0$, $e^{-x}=\tfrac{1}{2\pi i}\int_{\sigma=\sigma_0}x^{-s}\Gamma(s)ds$ where $s=\sigma+it$, and the integral is taken on a vertical line with fixed real part $\sigma_0>0$, and $-\infty<t<\infty$. [Hint: what is the residue of $x^{-s}\Gamma(s)$ at $s=-n$? Also, use problem (3) and the Stirling formula.]
\item Finish the proof that $\xi(s)=\xi(1-s)$ from Thm.~2 of Lecture 17.
\end{enumerate}

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