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1

It's very easy to give analytic continuations for $f(s) = \displaystyle \sum_{n \geq 1}\dfrac{a(n)}{n^s}$ when $a(n)$ is a periodic function (for example, $i^n$ of $\zeta^n$ for $\zeta$ a root of unity). Suppose the coefficients are periodic with period $Q$. Then write \begin{align} f(s) &= \sum_{0 \leq c < Q}a(c)\sum_{n \geq 1} \frac{1}{(c + ...

3

Hint: $e^{ikx}+e^{i(-k)x}=2cos(kx)$, for all $1\le k\le n$.

1

Every Dirichlet character $\chi$ (mod $5$) is determined by the value $\chi(2)$, since $(2,2^2,2^3,2^4) \equiv (2,4,3,1) \pmod 5$ and therefore $(\chi(1),\chi(2),\chi(3),\chi(4)) = (\chi(2)^4, \chi(2), \chi(2)^3, \chi(2)^2)$. (We're using the fact that $2$ is a "primitive root" modulo $5$.) On the other hand, we already know that $\chi(1)=1$. Therefore ...

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