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The Riemann Hypothesis, explained (2016) (medium.com)
119 points by psvidler 8 months ago | hide | past | web | favorite | 12 comments

Prime Obsession, by John Derbyshire, is a very entertaining and well written book about this topic, following a similar path (although much slower) among the conceptual building blocks required to understand the hypothesis.

John Derbyshire has become a controversial figure:


Wow, that's out there. It's such a shame that so many people whose work would otherwise be a net positive have to stain their reputations by being such outright a-holes in other aspects of life.

See https://news.ycombinator.com/item?id=13344071 for some interesting past discussion

> The fourth and final term is an integral which is zero for x < 2 because there are no primes smaller than 2.

This does not seem correct. Overall a good article though.

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

Of course. That was not the error.

Yeah, that part is wrong. It's a continuous function of x, so if its value at x=2 is 0.1400101..., it couldn't suddenly be 0 at every x < 2.

Note to other commenters: this has nothing to do with whether or not 1 is considered a prime, as the integral makes no reference to primes at all. OP was taking issue with the former claim in the quote, not the latter (and the latter, while true, appears to have nothing to say about the former; my guess is some kind of editing mistake).

Thanks. I was very brief because I was on my cell phone. Did not expect downvotes for that comment. :-)

I agree. This integral cannot be constantly zero in that region, as the integrand is not itself constantly zero.

Also, there's no reason to expect this integral (merely the last term of the explicit formula, the one arising from the trivial zeros of the zeta function) to count primes all in its own, so no reason it should be zero on its own below 2. That statement instead correctly describes the sum of all four terms, in the article's presentation.

I was doing some reading about the Rieman hypotesis like a month ago - when searching I came across this very same post - not sure if here or somewhere else, but I read that the stric definition of a prime number (It has to be divisible only by itself, and obviously _also_ by 1) rules out 1 bcause it is only divisible by itself. I mean, it is divisible by one, but that doesn't count as _also_.

Like everything else in mathematics, the definition of 'prime' which excludes 1 is a convenient convention. If we don't let 1 be a prime number, then by the "Fundamental theorem of arithmetic"[1] we can say that every number is either prime or a unique product of primes. If we let 1 be a prime, we could write e.g. 6=2x3, or 6=2x3x1x1, so the factorization would no longer be a unique product of primes.


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