**Problem 10. (16 points)** If

\[\sum_{a=1}^\infty\sum_{b=1}^\infty\frac{1}{a^4+4b^4}=\frac{\pi^p}{q},\]

then evaluate the integer value of \(p+q\).

**Announcement.** This is the last CMC problem of the season. I expect to return in about two weeks.

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TopNewest\(p = 4, q = 288 \Rightarrow p+q=292\)

I don't have a full answer because I couldn't prove one crucial step.

Because (I can't prove this) \( \displaystyle \sum_{a=1}^\infty \frac {8b^4}{a^4+4b^4} = b \pi \coth (b \pi) - 1 \)

\[ \begin{eqnarray} \displaystyle \sum_{a=1}^\infty \sum_{b=1}^\infty \frac {1}{a^4+4b^4} & = & \sum_{b=1}^\infty \frac {1}{8b^4} (b \pi \coth (b \pi) - 1 ) \nonumber \\ & = & \frac {\pi}{8} \left ( \displaystyle \sum_{b=1}^\infty \frac { \coth (b \pi) }{b^3} \right ) - \frac {1}{8} \left ( \sum_{b=1}^\infty \frac {1}{b^4} \right ) \\ & = & \frac {\pi}{8} \left ( \displaystyle \sum_{b=1}^\infty \frac { \frac {e^{2b \pi} + 1 }{e^{2b \pi} - 1 } }{b^3} \right ) - \frac {1}{8} \left ( \sum_{b=1}^\infty \frac {1}{b^4} \right ) \\ & = & \frac {\pi}{8} \left ( \displaystyle \sum_{b=1}^\infty \frac {2 + e^{2b \pi} -1 }{b^3( e^{2b \pi} - 1 )} \right ) - \frac {1}{8} \left ( \sum_{b=1}^\infty \frac {1}{b^4} \right ) \\ & = & \frac {\pi}{8} \left ( \displaystyle \sum_{b=1}^\infty \frac {1}{b^3} + \sum_{b=1}^\infty \frac {2}{b^3( e^{2b \pi} - 1 )} \right ) - \frac {1}{8} \zeta (4) \\ & = & \frac {\pi}{8} \left ( \zeta (3) + \frac {7 \pi^3}{180} - \zeta (3) \right ) - \frac {1}{8} \zeta (4) \\ & = & \frac {\pi}{8} \left ( \frac {7 \pi^3}{180} - \frac {\pi^3}{90} \right ) = \frac {\pi^4}{288} \\ \end {eqnarray} \]

Note: from this, \( \displaystyle \zeta (3) = \frac {7 \pi^3}{180} - 2 \sum_{n=1}^\infty \frac {1}{n^3( e^{2n \pi} - 1 )} \), and \( \zeta (4) = \frac {\pi^4}{90} \)

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A completion of the starting step with the Poisson summation formula:

Fix \(b\) and consider \[f(a) = \frac{1}{a^4 + 4b^4}.\] First, we compute the (continuous) Fourier transform \[ \hat{f}(c) = \int_{-\infty}^{\infty} e^{-2i\pi ct}\cdot \frac{1}{t^4 + 4b^4}\,dt \] with a contour integral.

Denote the integrand as \(g(t)\), i.e. \[g(t) = e^{-2i\pi ct}\cdot \frac{1}{t^4 + 4b^4}.\]

Suppose \(c \geq 0\). Then if \(\Im(t) \leq 0\), we have \(\Re(-2i\pi ct) \leq 0\) and \(|e^{-2i\pi ct}| \leq 1\). Consider the contour integral of \(g\) around a large semicircle centered at the origin in the half-plane \(\Im(t) \leq 0\) of the complex plane. Since \(\frac{1}{t^4 + 4b^4}\) decays quickly enough if \(t\) has large magnitude, if the semicircle's radius goes to infinity, the integral over the arc of the semicircle goes to 0, so that the contour integral converges to the integral over the real line, \[ \int_{\infty}^{-\infty} g(t)\,dt. \] (Note the orientation, since we need to go around the semicircle counterclockwise.)

At the same time, the integral can be evaluated with the residue formula. Let \(\omega = e^{i\pi/4}\), a primitive eighth root of unity. \(g\) has four simple poles at \(t = \sqrt{2}\omega^k b\) for \(k = 1, 3, 5, 7\); the relevant poles inside our contour are \(\sqrt{2}\omega^5b\) and \(\sqrt{2}\omega^7b\). The residues at those points can be evaluated in an L'Hôpital-esque manner to be:

\[ \begin{align*} \operatorname{res}_{\sqrt{2}\omega^5b} g &= e^{-2i\pi c(\sqrt{2}\omega^5b)}\cdot \frac{1}{4(\sqrt{2}\omega^5b)^3} \\ &= e^{2\pi bc(-1+i)}\cdot \frac{1+i}{16b^3} \\ \operatorname{res}_{\sqrt{2}\omega^7b} g &= e^{-2i\pi c(\sqrt{2}\omega^7b)}\cdot \frac{1}{4(\sqrt{2}\omega^7b)^3} \\ &= e^{2\pi bc(-1-i)}\cdot \frac{-1+i}{16b^3} \\ \end{align*} \]

Additionally supposing \(c\) is an integer, we then have \(e^{2\pi bci} = 1\) and can further simplify to \[ \begin{align*} \operatorname{res}_{\sqrt{2}\omega^5b} g &= e^{-2\pi bc}\cdot \frac{1+i}{16b^3} \\ \operatorname{res}_{\sqrt{2}\omega^7b} g &= e^{-2\pi bc}\cdot \frac{-1+i}{16b^3} \end{align*} \]

So, \[ \begin{align*} \int_{\infty}^{-\infty} g(t)\,dt &= 2\pi i\left( e^{-2\pi bc}\cdot \frac{1+i}{16b^3} + e^{-2\pi bc}\cdot \frac{-1+i}{16b^3}\right) \\ &= -\frac{\pi}{4b^3} \cdot e^{-2\pi bc} \\ \hat{f}(c) &= \frac{\pi}{4b^3} \cdot e^{-2\pi bc} \\ \end{align*} \]

Note that since \(f\) is even, \(\hat{f}(c) = \hat{f}(-c)\). Now the Poisson summation formula can be evaluated as two geometric series.

\[ \sum_{a=-\infty}^\infty f(a) = \sum_{c=-\infty}^\infty \hat{f}(c) = \frac{\pi}{4b^3} \left(\frac{1}{1 - e^{-2\pi b}} + \frac{e^{-2\pi b}}{1 - e^{-2\pi b}}\right) = \frac{\pi \coth(b\pi)}{4b^3} \]

Thus, again since \(f\) is even, \[ \begin{align*} \sum_{a=1}^\infty f(a) &= \frac{1}{2}\left(\sum_{a=-\infty}^\infty f(a) - f(0)\right) \\ &= \frac{\pi \coth(b\pi)}{8b^3} - \frac{1}{8b^4} \\ &= \frac{1}{8b^4}\left(b\pi \coth(b\pi) - 1\right), \end{align*} \] as desired.

I have no idea how the parent post managed to continue from here, or how Sophie-Germain can help, however.

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mind = blown, 8 points for you

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Now, how do you solve it using Sophie-Germain Identity?

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If it's better than Brian Chen's answer, that is if you don't need to use Poisson summation formula, I would like to know...

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Excellent progress, 7 points. Can you finish the proof using the hint?

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Has anyone proven the first step? If not, I'd like to take a shot.

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Do I really need the knowledge of Fourier to really understand or maybe Riemann Zeta Function?

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how do you write the solution

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Hint: this problem is related to this problem.

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You should probably clear up any ambiguity relating to the fact that you used \(a, b\) as dummy variables and also in the answer form.

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Fixed.

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Hint #2: Sophie-Germain Identity

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I'm still waiting for an explanation of how this helps, and I bet I'm not the only one ;-)

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See my comment under Pi Han Goh's comment.

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cody i wanna ask you that how can i submit my solution .... using all the maths terms??? please help

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Just post your solution to this thread.

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what is this equation

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I've always wondered this, what does the notation mean when you have two sigmas next to each other? Does this mean that they both start at 1, then both go to 2, then both go to 3? Or does it mean the sum of a = 1 and b = 1 to infinity, the sum of a = 2 and b = 1 to infinity, the sum of a = 3 and b =1 to infinity, etc..?

From this problem I think it's the latter or else it would just be stated as "\( \displaystyle \sum_{a = 1}^{\infty} \frac{1}{5a^4}\).

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\[\displaystyle \sum_{a=1}^m \displaystyle\sum_{b=1}^n f(a, b)\] means \[\displaystyle \sum_{a=1}^m \left(\displaystyle\sum_{b=1}^n f(a, b)\right),\] or \[(f(1, 1) + f(1, 2) + \ldots + f(1, n)) \\+ (f(2, 1) + f(2, 2) + \ldots + f(2, n)) \\+ \ldots \\ + (f(m, 1) + f(m, 2) + \ldots + f(m, n)).\]

In other words, we evaluate the inner sums first, taking outside variables as constant.

Note that (in this example at least) we could

switchthe order of the summation symbols, to get \[\displaystyle \sum_{b=1}^n \displaystyle\sum_{a=1}^m f(a, b)\] or \[(f(1, 1) + f(2, 1) + \ldots + f(m, 1)) \\+ (f(1, 2) + f(2, 2) + \ldots + f(m, 2)) \\+ \ldots \\ + (f(1, n) + f(2, n) + \ldots + f(m, n)).\]See how these two are actually the same?

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Be careful with limits to infinity. You cannot just interchange the order of summation, as that can affect the sum itself. Analysis deals with this, and one of the results is that if the sequence converges absolutely to a finite value, then we can rearrange the terms and it will still converge to the same (finite) value.

For fun, take the sequence \( - \frac{1}{i} \), and rearrange terms to get get it to converge to any value that you wish. The absolute value of this sequence is the harmonic sequence \( \frac{1}{i}\), which sums to infinity.

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And if you're a computer-sciency type of person, the first way to write it is the same as

and the second way is the same as

which both return the same number.

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\(p + q = 292\)

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I'll award 2 points for this answer, considering the magnitude of this problem. But the solution's where it's at.

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