(Spoilers!) Comments and discussion on Valentin's 5th "Then and Now" - Roots

[EdS2]

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.... possible spoiler thoughts follow ...
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First off, the article Valentin mentions

Quote: Some of you might remember an article where I computed an approximation to the ∏(x) function, which gives the number of primes up to some given limit

is surely VA027 ("Small Fry - Primes A'counting") found on his articles page.

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We can read more about Riemann's prime-counting function R(x) on Wikipedia here and in Wolfram's MathWorld here. Warning: at that second link you will find some information about the roots, which are the object of the challenge.

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On this page you'll see more about this R(x) function, and in particular there's an animation showing how a smooth function progressively approximates the prime-counting stepwise function as more terms in the series are added.

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I think this might be a clue: we have a wiggly function, and near the origin we might wonder if those wiggles take the function negative.

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Perhaps by studying the function we can figure out how small the argument might need to be, and figure out how many terms we'll need to be convincing about the locations of the zeros, if we find them.

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Perhaps the final code will look a little bit like the code found in VA027...