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>> But of course, the primes aren’t random. Stunning to me that a staff writer for a science magazine would type this sentence without referencing the riemann
by reedf1 2y ago
>> But of course, the primes aren’t random.
Stunning to me that a staff writer for a science magazine would type this sentence without referencing the riemann hypothesis.
- lblume 2y agoQuanta Magazine tries to make these topics accessible for people with low knowledge regarding them, thus mentioning the Riemann hypothesis, although adequate and needed for, say, a lecture on the topic, would not really help this goal.
- griffzhowl 2y agoBut the primes are obviously not random, independently of the Riemann hypothesis - they're determinate consequences of the number system. Maybe I missed your point
- graycat 2y agoIn 1933, A. Kolmogorov used H. Lebesgue's measure theory to define random variables. With that definition, could have a random variable X whose values are only prime numbers and such that for each prime number p and for probability measure P, P(X = p > 0), that is, the probability that X = p is positive. So, with X, the prime numbers are random. References: With TeX markup, polished details on measure theory are in H.\ L.\ Royden, {\it Real Analysis: Second Edition,\/} Macmillan, New York, 1971.\ \ Walter Rudin, {\it Real and Complex Analysis,\/} ISBN 07-054232-5, McGraw-Hill, New York, 1966.\ \ and polished details on probability theory based on measure theory are in Leo Breiman, {\it Probability,\/} ISBN 0-89871-296-3, SIAM, Philadelphia, 1992.\ \ Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability,\/} Holden-Day, San Francisco, 1965.\ \
- ecmm 2y agoWhat does "with X" mean in this context?
- graycat 2y agoUh, Joe, you have random variable X. What is its value? Sam, just a minute. Let me draw a sample. Got one: X = 7. But, Joe 7 is not very surprising or interesting. Is there anything else? Sam, sure, one more minute. Got one! Joe, well, then, WHAT is it???? Sam, sorry, it has 2^12345 digits and will take a while to print it out; this morning I have a coffee shop meeting with Susan; and I don't want to miss Susan, cute, pretty, sweet, smart, darling, adorable, precious ..., and single! Exercise: Show that there is a random variable Y with the same distribution as X and such that X and Y are independent random variables. I.e., knowledge of X tells us nothing about Y. The intuitive concept of random is closer to unpredictable given even everything else, that is, what probability theory defines as independent. There are more details in Royden, Rudin, Breiman, and Neveu. To preview, there is a non-empty set Omega with a collection F of subsets that form a sigma algebra and a measure P on F. Then random variable X is a measurable function from Omega to the set of all prime numbers. So, for some point w in Omega and function X, X(w) is a prime number. Can think of w as a trial. Uh, this morning I'm working on some Rexx code, so here I can't reproduce or compete with the references by Royden, Rudin, Breiman, and Neveu.
- graycat 2y agoEdit: Replace P(X = p > 0) with P(X = p) > 0
- ecmm 2y agohttps://en.wikipedia.org/wiki/Riemann_hypothesis?oldformat=true#Distribution_of_prime_numbers https://en.wikipedia.org/wiki/Riemann_hypothesis?oldformat=t...
- wruza 2y agooldformat=true I’m curious what this does, cause it seemingly changes nothing (tested on mobile, with “desktop version” too).
- YetAnotherNick 2y agoRiemann hypothesis(conjecture) doesn't prove much for prime, other than a tight bound on the prime counting function. Most just mean related Riemann's explicit formula for prime when they link Riemann hypothesis and prime number.
- bmacho 2y agoWeird. It is obviously true. Also it is redundantly explained in the very next half sentence: > But of course, the primes aren’t random. They’re completely determined, Also this whole thread that this post started is so stupid.