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Interesting puzzle here for someone good at combinatorics and asymptotics. Any takers?

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Oct 22, 2021

Replying to @stevenstrogatz

Steven I was quite awake last night because of this problem. I wrote it down nicely here gist.github.com/buddhabrot/2e3 Could you reach out to anyone who can help with demystifying this? It's a little scary because the constant factor does not seem to make sense. cc @johncarlosbaez

11:58 AM · Oct 22, 2021

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The right hand side looks like a Riemann sum. Maybe that will help.

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Thanks for sharing, I hope this meets a happy ending. By the way if the conjecture is true, that means there are no more integer solutions for the form (since C is irrational).

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Wait. Your numerical solution is not so close to 1/sqrt(5). One of them is ~.4472 and the other is ~.4427

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Start by replacing x by the new variable u=x/n, then approximate the terms of the sum by the corresponding exponentials using Euler's formula. You get a geometric series that you can sum, and from there get the asymptotic of the root (my 5 min thought. I haven't checked details).

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Unrelated Question but how do I find the partial sums of Polygamma (-√(x^2 + 356x + 321)) ?, I know I can resolve negative Polygamma by reflection to positive Polygamma and πcot(π(x^2 + 356x + 321)). So I guess I really need to know the partial sums over cot(x).