New insane limit!

Calculus Level 4

For R > 1 R>1 , let D R = { ( a , b ) ∈ Z 2 : 0 < a 2 + b 2 < R } \mathcal{D}_R =\{ (a,b)\in \mathbb{Z}^2: 0<a^2+b^2<R\} .

Compute lim ⁡ R → ∞ ∑ ( a , b ) ∈ D R ( − 1 ) a + b a 2 + b 2 . \displaystyle \lim_{R\rightarrow \infty}{\sum_{(a,b)\in \mathcal{D}_R}{\frac{(-1)^{a+b}}{a^2+b^2}}}.

Your answer comes in the form - ( π ) l o g m (\pi)logm .

Find m m .

Thanks to Jon Haussmann for pointing out mistake....


The answer is 2.

Rajyawardhan Singh by Rajyawardhan Singh , 19, India

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1 solution

Rohan Shinde
Mar 12, 2019

May I ask the source of this question? Amazing question indeed. Had to use Gaussian integers and the Dirichlet L-series (Along with its relation to the Dirichlet lambda function and Dirichlet Beta function) to find the answer. But was worth it.

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