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I'm havig difficulty with solving an exercise involving the computation of an Eisenstein series.

In this exercise we consider the cotangent function $\text{cot(z)}=\frac{\cos(z)}{\sin(z)}$.

a) Show that $\text{cot}$ has simple poles at all integer multiples of $\pi$ and is defined on all other points $\mathbb{C} \setminus \pi \mathbb{Z}$.

b) For fixed $z_0 \in \mathbb{C} \setminus \mathbb{Z}$ consider the function

$$ f(z)=\frac{\pi \text{cot}( \pi z)}{z_0-z}. $$ Calcute the residues of $f$ at all of its singularities. (Hint: Use $\underset{z=z_0}{ \text{res}} \ f(z)=\lim_{z \to z_0} (z-z_0) f(z)$).

c) Show that $ \lim_{N \to \infty} \oint_{|z|=N+\frac{1}{2}} f(z)dz=0$. (Hint: Consider the integral $\oint_{|z|=N+ \frac{1}{2}} z^{-1} \pi \text{cot}(\pi z) dz$.)

d) Use your results to calculate, for $z \in \mathbb{C} \setminus \mathbb{Z}$,

$$ \lim_{N \to \infty} \sum_{n=-N}^{N} \frac{1}{z-n}. $$

e) (Bonus) Use the methods from the problems above to give a proof that

$$ \zeta(2)=\frac{\pi^2}{6}. $$

I have problems with c) and d). I considered the integrals

$$ \oint_{|z|=N+\frac{1}{2}} \frac{\pi \text{cot}( \pi z)}{z_0-z} dz \tag{1} \\ $$

$$ \oint_{|z|=N+\frac{1}{2}} \frac{\pi \text{cot}( \pi z)}{z} dz \tag{2}. $$

I know from b) that the residues of $f$ are

$$ \underset{z=z_0}{\text{res}}f(z)=- \pi \text{cot}(\pi z_0). $$

$$ \underset{z=k \pi}{\text{res}}f(z)=\frac{1}{z_0-k}, \ \ k \in \mathbb{Z}. $$

Moreover, it is clear that the integrand of (2) is almost the integrand of (1) (to see this set $z_0=0$ in the integrand of (1) and multiply by (-1)) and threfore the two integrals are almost equal. Given that I know the residues of $f$ I think I am supposed to use the residue theorem to compute (2) and then use the result to compute (1). But I do not see how to exactly proceed. I also do not see how to solve d) with help of the previous results. The only thing I see is that the denominator of $\frac{1}{z-n}$ resembles the denominator of $f$ so I expect the series will translate into an expression involving the integral computed in c). But so far I have not found an explicit strategy.

\Edit: Corrected the result for the residues on the real line.

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First of all, the series $\sum_{k\in} \dfrac{1}{z-k}$ does not converge: you have to rearrange the terms ($k$ with $-k$) to have a convergent form.

Then, show that:

  • your two functions are meromorphic
  • they have the same pole
  • the pole are of same degree
  • the difference is a constant
  • compute that constant with a suitable limit (I think you should look at the limit while $t\to \infty$ of the difference evaluated in $it$)

Thus, you have two different writings of the same function. One would give you $\zeta(2)$, the other $\frac{\pi^2}{6}$.

Edit I forgot to say what are the two functions. The one is the series and the other is $z \mapsto \pi\mathrm{cotan} (\pi z)$.

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