Questions tagged [poisson-summation-formula]

For questions dealing with the Poisson Summation Formula

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2
votes
2answers
112 views

Evaluating $\sum_{k=1}^{\infty}\left(\frac{\sin(tk)}{k}\right)^2$

Using Poisson Summation Formula, how do you evaluate the following infinite sum $\sum_{k=1}^{\infty}\left(\frac{\sin(tk)}{k}\right)^2$? The Poisson Summation Formula states that: $\sum_{k=-\infty}^{\...
21
votes
3answers
513 views

A Ramanujan sum involving $\sinh$

Today, in a personal communication, I was asked to prove the classical result $$\boxed{ \sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\sinh(\pi n)} = \frac{\pi^3}{360}}\tag{CR} $$ which I believe is due to ...
3
votes
1answer
69 views

Probability density of the fractional part of |σZ| as σ→∞?

Consider the following probability density function parameterized by $\sigma>0$: $$g_\sigma(x)=2\sum_\limits{n=0}^\infty\phi_\sigma(x+n), \quad 0<x<1$$ where $$\phi_\sigma(t)={1\over \sigma\...
2
votes
1answer
207 views

Proving the Partial Fraction Decomposition of the Hyperbolic Cotangent Function by using Poisson Summation

While skimming through the wonderful post What are some examples of colorful language in serious mathematics papers? on MathOverflow an example given by Ben Green aroused my curiosity. He referred to ...