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## 3. Another section

testing $x^x$ foo bar
1.     Let $c(h/k):=V(\overline{h},k).$

#### Problem 3.05.

[Brian Conrey] Prove that $c$ satisfies a reciprocity formula \begin{eqnarray} xc(x)+c(1/x)-\frac{1}{\mbox{Den}(x)}=\psi(x) \end{eqnarray} where $\psi(x)$ can be analytically continued to $\mathbf{C}'$, the complex plane minus the negative real axis.
1. Remark. What does "Den" mean?
•     This is a complicated equation

#### Problem 3.15.

\begin{eqnarray*} \sum_{n=1}^\infty \frac{1}{n^2}&=& 1+\frac{1}{2^2}+\frac{1}{3^2}+\dots \\ &=& \frac{\pi^2}{6}. \end{eqnarray*}
• ### The Four Color theorem

This is a famous old problem.

#### Conjecture 3.2.

[Rob Beezer] A planar map can be colored with four or fewer colors and there is a proof without a laborious number of cases.
Famously not yet done.
1. Remark. Test remark
• Remark. 33
• Remark. [org.aimpl.user:aaa] Something new.
• Remark. testo
•     Recall that the Vasyunin sum is $$V(h,k)=\sum_{a=1}^{k-1}\frac{a}{k} \cot \frac{\pi \overline{a} h}{k}.$$

#### Problem 3.45.

[Brian] Prove that $$V(-h,k)=V(h,k).$$
Unsolved.